How do you use partial fraction decomposition to decompose the fraction to integrate #4/((x^2+9)(x+1))#?
See the explanation below
The decomposition into partial fractions is
The denominators are the same, compare the numerators
Therefore,
So, the integral is
Finally,
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To decompose the fraction 4/((x^2+9)(x+1)) using partial fraction decomposition, follow these steps:

Factor the denominator: x^2 + 9 can be factored as (x + 3i)(x  3i), where i is the imaginary unit.

Express the fraction in terms of partial fractions: 4/((x^2+9)(x+1)) = A/(x+1) + (Bx + C)/(x^2+9)

Clear the fractions by multiplying both sides by the common denominator (x^2+9)(x+1).

Solve for A, B, and C by comparing coefficients of like terms.

Once you find the values of A, B, and C, integrate each term separately.
The steps may involve algebraic manipulations and solving a system of equations to find the unknown constants A, B, and C. Once you have the partial fraction decomposition, you can integrate each term separately to find the integral of the original expression.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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