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

This is one hell of a process for a seemingly simple integral..

Note that we need the numerator to be a binomial since both the denominators are polynomials of degree two.

The process for evaluating these two integrals is basically the same, so I'm only going to show the left one.

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To integrate 1/(x^4 + 1) using partial fractions, we first factor the denominator as (x^2 + 1)(x^2 - 1). Then, we express 1/(x^4 + 1) as the sum of two partial fractions:

1/(x^4 + 1) = A/(x^2 + 1) + B/(x^2 - 1)

Now, we need to find the values of A and B. We can do this by multiplying both sides of the equation by (x^2 + 1)(x^2 - 1) to clear the denominators:

1 = A(x^2 - 1) + B(x^2 + 1)

Expanding and combining like terms, we get:

1 = (A + B)x^2 - (A + B)

Now, we equate the coefficients of like powers of x:

For x^2 term: 0 = A + B For constant term: 1 = -A - B

From the first equation, we get A = -B. Substituting this into the second equation:

1 = -A - A 1 = -2A A = -1/2

Since A = -B, B = 1/2.

Therefore, the partial fraction decomposition is:

1/(x^4 + 1) = -1/2(x^2 + 1) + 1/2(x^2 - 1)

Now, we can integrate each partial fraction separately:

∫(-1/2)(x^2 + 1) dx + ∫(1/2)(x^2 - 1) dx

= -1/2∫(x^2 + 1) dx + 1/2∫(x^2 - 1) dx

= -1/2(1/3)x^3 - 1/2x + 1/2(1/3)x^3 - 1/2x + C

= -(1/6)x^3 - x - (1/6)x^3 + x + C

= -(1/3)x^3 + C

So, the integral of 1/(x^4 + 1) using partial fractions is -(1/3)x^3 + C.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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