# Math virtual assistant

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# Step-by-step Solution

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## Answer

$3x-\ln\left|x+1\right|-2\ln\left|x+2\right|+C_0$

## Step-by-step explanation

Problem to solve:

$\int\frac{3x^2+6x+2}{x^2+3x+2}dx$
1

Divide $3x^2+6x+2$ by $x^2+3x+2$

$\begin{array}{l}\phantom{\phantom{;}x^{2}+3x\phantom{;}+2;}{\phantom{;}3\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+3x\phantom{;}+2\overline{\smash{)}\phantom{;}3x^{2}+6x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+3x\phantom{;}+2;}\underline{-3x^{2}-9x\phantom{;}-6\phantom{;}\phantom{;}}\\\phantom{-3x^{2}-9x\phantom{;}-6\phantom{;}\phantom{;};}-3x\phantom{;}-4\phantom{;}\phantom{;}\\\end{array}$
2

Resulting polynomial

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

## Answer

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

### Main topic:

Integrals by partial fraction expansion

~ 1.47 seconds