# Step-by-step Solution

## Integral of $\frac{5x^2+14x+10}{\left(x+1\right)^2\left(x+2\right)}$ with respect to x

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

## Step-by-step explanation

Problem to solve:

$\int\frac{5x^2+14x+10}{\left(x+2\right)\left(x+1\right)^2}dx$
1

Rewrite the fraction $\frac{5x^2+14x+10}{\left(x+1\right)^2\left(x+2\right)}$ in $3$ simpler fractions using partial fraction decomposition

$\frac{5x^2+14x+10}{\left(x+1\right)^2\left(x+2\right)}=\frac{A}{x+2}+\frac{B}{\left(x+1\right)^2}+\frac{C}{x+1}$
2

Find the values of the unknown coefficients. The first step is to multiply both sides of the equation by $\left(x+1\right)^2\left(x+2\right)$

$5x^2+14x+10=\left(x+1\right)^2\left(x+2\right)\left(\frac{A}{x+2}+\frac{B}{\left(x+1\right)^2}+\frac{C}{x+1}\right)$

$2\ln\left|x+2\right|+\frac{-1}{x+1}+3\ln\left|x+1\right|+C_0$
$\int\frac{5x^2+14x+10}{\left(x+2\right)\left(x+1\right)^2}dx$