How do you integrate #1 / (x(x^4+1))# using partial fractions?

Perform the decomposition into partial fractions

The denominators are the same, compare the numerators

Therefore,

So,

Perform the second part by substitution

So,

Finally,

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[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

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1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

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[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

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1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fractionTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, weTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction intoTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express (To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partialTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractionsTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \fracTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{xTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(xTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 /To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(xTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)}To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 +To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} \To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} )To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) asTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1))To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sumTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) =To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partialTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / xTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractionsTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x +To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (BTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \fracTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (BxTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx +To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + CTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) /To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{xTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(xTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (xTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 +To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)}To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} =To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1)To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \fracTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (DTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx +To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x}To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E)To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (xTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{BTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{BxTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx +To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 -To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + CTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{xTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3.To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. ClearTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 +To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear theTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractionsTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1}To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions byTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} +To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplyingTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying bothTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \fracTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sidesTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx +To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + ETo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{xTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 =To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = ATo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 -To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 +To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1}To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

MultipTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

MultiplyingTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(xTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying throughTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x -To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through byTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by theTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominatorTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1)To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator (To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) +To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( xTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(xTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (BTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (BxTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx +To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 +To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + CTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)xTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x -To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1)To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ),To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), weTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1)To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we getTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) +To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (DxTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx +To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + ETo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 =To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)xTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(xTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(xTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, CTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C,To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + CTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, DTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D,To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E byTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparingTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficientsTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1)To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of likeTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) +To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like termsTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms onTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (DTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on bothTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx +To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sidesTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x +To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5.To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. AfterTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1)To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solvingTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) \To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A,To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

NowTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, BTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now,To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B,To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, CTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equateTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, DTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients ofTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, andTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like termsTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and ETo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms onTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrateTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sidesTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for (To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( ATo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. TheTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A,To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integralTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, BTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral ofTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B,To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of eachTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C,To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each termTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, DTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term willTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D,To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will giveTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and (To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the finalTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( ETo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E \To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

TheTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral ofTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After findingTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding theTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (xTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the valuesTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(xTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of (To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( ATo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A,To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4 +To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A, BTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4 + To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A, B,To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4 + 1))To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A, B, CTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4 + 1)) usingTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A, B, C,To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4 + 1)) using partialTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A, B, C, D,To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4 + 1)) using partial fractionsTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A, B, C, D, \To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4 + 1)) using partial fractions will involveTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A, B, C, D, )To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4 + 1)) using partial fractions will involve logarithTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A, B, C, D, ) andTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4 + 1)) using partial fractions will involve logarithmicTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A, B, C, D, ) and (To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4 + 1)) using partial fractions will involve logarithmic andTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A, B, C, D, ) and ( E \To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4 + 1)) using partial fractions will involve logarithmic and inverseTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A, B, C, D, ) and ( E ),To integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4 + 1)) using partial fractions will involve logarithmic and inverse tangentTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A, B, C, D, ) and ( E ), theTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4 + 1)) using partial fractions will involve logarithmic and inverse tangent functionsTo integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A, B, C, D, ) and ( E ), the integralTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4 + 1)) using partial fractions will involve logarithmic and inverse tangent functions.To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A, B, C, D, ) and ( E ), the integral ofTo integrate 1 / (x(x^4+1)) using partial fractions, follow these steps:

1. Factor the denominator: x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1)

2. Decompose the fraction into partial fractions: 1 / (x(x^4 + 1)) = A / x + (Bx + C) / (x^2 + 1) + (Dx + E) / (x^2 - 1)

3. Clear the fractions by multiplying both sides by the denominator: 1 = A(x^2 + 1)(x - 1) + (Bx + C)x(x - 1) + (Dx + E)x(x + 1)

4. Solve for the unknown coefficients A, B, C, D, and E by comparing coefficients of like terms on both sides of the equation.

5. After solving for A, B, C, D, and E, integrate each term separately.

6. The integral of each term will give the final result.

The integral of 1 / (x(x^4 + 1)) using partial fractions will involve logarithmic and inverse tangent functions.To integrate ( \frac{1}{x(x^4+1)} ) using partial fractions, first factorize the denominator.

[ x(x^4 + 1) = x(x^2 + 1)(x^2 - 1) = x(x^2 + 1)(x + 1)(x - 1) ]

Then, we express ( \frac{1}{x(x^4+1)} ) as a sum of partial fractions:

[ \frac{1}{x(x^4+1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} ]

Multiplying through by the denominator ( x(x^4 + 1) ), we get:

[ 1 = A(x^2 + 1)(x - 1) + (Bx + C)(x)(x - 1) + (Dx + E)(x)(x + 1) ]

Now, equate coefficients of like terms on both sides and solve for ( A, B, C, D, ) and ( E ).

After finding the values of ( A, B, C, D, ) and ( E ), the integral of ( \frac{1}{x(x^4+1)} ) can be expressed as:

[ \int \frac{1}{x(x^4+1)} , dx = \int \left( \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - 1} \right) , dx ]

[ = A \ln|x| + B \arctan(x) + \frac{C}{2} \ln(x^2 + 1) + \frac{1}{2} \left( D \ln|x^2 - 1| + E \ln|x^2 - 1| \right) + C_1 ]