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Step-by-step Solution

Integral of $\frac{4x^2+2x+8}{x\left(x^2+2\right)^2}$ with respect to x

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Answer

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

Step-by-step explanation

Problem to solve:

$\int\frac{4x^2+2x+8}{x\left(x^2+2\right)^2}$
1

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

$\frac{4x^2+2x+8}{x\left(x^2+2\right)^2}=\frac{A}{x}+\frac{Bx+C}{\left(x^2+2\right)^2}+\frac{Dx+F}{x^2+2}$
2

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

$4x^2+2x+8=x\left(x^2+2\right)^2\left(\frac{A}{x}+\frac{Bx+C}{\left(x^2+2\right)^2}+\frac{Dx+F}{x^2+2}\right)$

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Answer

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