Question

Solve: $\int \frac{x+1}{\left(x+2\right)\left(x+3\right)}dx$

Answer 1

$\int \frac{x+1}{\left(x+2\right)\left(x+3\right)}dx$
integrating by partial fraction
$\begin{array}{c}\frac{x+1}{\left(x+2\right)\left(x+3\right)}=\frac{A}{\left(x+2\right)}+\frac{B}{\left(x+3\right)}\\ \left(x+1\right)=A\left(x+3\right)+B\left(x+2\right)\\ x+1=\left(A+B\right)x+3A+2B\end{array}$
comparing both sides,
$\begin{array}{c}A+B=1\\ 3A+2B=1\\ so,A=-1,B=2\end{array}$
$I=\int \frac{-1}{\left(x+2\right)}dx+\int \frac{2}{\left(x+3\right)}dx$

$=-\log (x+2)+2\log (x+3)$