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Step-by-step Solution

Integral of $\frac{-x^2+8x^2-9x+2}{\left(x-3\right)^2\left(x^2+1\right)}$ with respect to x

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Answer

$-\frac{51}{100}\ln\left|x^2+1\right|+\frac{7}{50}arctan\left(x\right)+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|+C_0$

Step-by-step explanation

Problem to solve:

$\int\:\frac{-x^2+8x^2-9x+2}{\left(x^2+1\right)\left(x-3\right)^2}dx$
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Adding $-1x^2$ and $8x^2$

$\int\frac{7x^2-9x+2}{\left(x-3\right)^2\left(x^2+1\right)}dx$
2

Rewrite the fraction $\frac{7x^2-9x+2}{\left(x-3\right)^2\left(x^2+1\right)}$ in $3$ simpler fractions using partial fraction decomposition

$\frac{7x^2-9x+2}{\left(x-3\right)^2\left(x^2+1\right)}=\frac{Ax+B}{x^2+1}+\frac{C}{\left(x-3\right)^2}+\frac{D}{x-3}$

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Answer

$-\frac{51}{100}\ln\left|x^2+1\right|+\frac{7}{50}arctan\left(x\right)+\frac{-\frac{19}{5}}{x-3}+\frac{51}{50}\ln\left|x-3\right|+C_0$