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Step-by-step Solution

Integral of (x^3-x^2+x-3)/(x+1)

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Answer

$\frac{x^{3}}{3}-x^2+3x-6\ln\left|x+1\right|+C_0$

Step-by-step explanation

Problem to solve:

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

Divide $x^3-x^2+x-3$ by $x+1$

$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}+1;}{\phantom{;}x^{2}-2x\phantom{;}+3\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}+1\overline{\smash{)}\phantom{;}x^{3}-x^{2}+x\phantom{;}-3\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}+1;}\underline{-x^{3}-x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{3}-x^{2};}-2x^{2}+x\phantom{;}-3\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+1-;x^n;}\underline{\phantom{;}2x^{2}+2x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}2x^{2}+2x\phantom{;}-;x^n;}\phantom{;}3x\phantom{;}-3\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+1-;x^n-;x^n;}\underline{-3x\phantom{;}-3\phantom{;}\phantom{;}}\\\phantom{;;-3x\phantom{;}-3\phantom{;}\phantom{;}-;x^n-;x^n;}-6\phantom{;}\phantom{;}\\\end{array}$
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Resulting polynomial

$\int\left(x^{2}-2x+3+\frac{-6}{x+1}\right)dx$

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Answer

$\frac{x^{3}}{3}-x^2+3x-6\ln\left|x+1\right|+C_0$