# Math virtual assistant

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# Step-by-step Solution

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## Answer

$4arctan\left(x\right)+\frac{4}{x-1}+\frac{-4}{\left(x-1\right)^{2}}+C_0$

## Step-by-step explanation

Problem to solve:

$\int\frac{8x+8}{\left(x-1\right)^3\left(x^2+1\right)}dx$
1

Rewrite the fraction $\frac{8x+8}{\left(x-1\right)^3\left(x^2+1\right)}$ in $4$ simpler fractions using partial fraction decomposition

$\frac{8x+8}{\left(x-1\right)^3\left(x^2+1\right)}=\frac{A}{\left(x-1\right)^3}+\frac{Bx+C}{x^2+1}+\frac{D}{x-1}+\frac{F}{\left(x-1\right)^{2}}$
2

Find the values of the unknown coefficients. The first step is to multiply both sides of the equation by $\left(x-1\right)^3\left(x^2+1\right)$

$8x+8=\left(x-1\right)^3\left(x^2+1\right)\left(\frac{A}{\left(x-1\right)^3}+\frac{Bx+C}{x^2+1}+\frac{D}{x-1}+\frac{F}{\left(x-1\right)^{2}}\right)$

## Answer

$4arctan\left(x\right)+\frac{4}{x-1}+\frac{-4}{\left(x-1\right)^{2}}+C_0$
$\int\frac{8x+8}{\left(x-1\right)^3\left(x^2+1\right)}dx$

### Main topic:

Integrals of Rational Functions

~ 0.98 seconds