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Step-by-step Solution

Integral of $\frac{x^3+2}{x^2-2x+2}$ with respect to x

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Answer

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

Step-by-step explanation

Problem to solve:

$\int\frac{x^3+2}{x^2-2x+2}dx$
1

Divide $x^3+2$ by $x^2-2x+2$

$\begin{array}{l}\phantom{\phantom{;}x^{2}-2x\phantom{;}+2;}{\phantom{;}x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{;}x^{2}-2x\phantom{;}+2\overline{\smash{)}\phantom{;}x^{3}\phantom{-;x^n}\phantom{-;x^n}+2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-2x\phantom{;}+2;}\underline{-x^{3}+2x^{2}-2x\phantom{;}\phantom{-;x^n}}\\\phantom{-x^{3}+2x^{2}-2x\phantom{;};}\phantom{;}2x^{2}-2x\phantom{;}+2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-2x\phantom{;}+2-;x^n;}\underline{-2x^{2}+4x\phantom{;}-4\phantom{;}\phantom{;}}\\\phantom{;-2x^{2}+4x\phantom{;}-4\phantom{;}\phantom{;}-;x^n;}\phantom{;}2x\phantom{;}-2\phantom{;}\phantom{;}\\\end{array}$
2

Resulting polynomial

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

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Answer

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