Question

Integrate with respect to $x$
$\frac{1}{(x+1)(x+2)}$

Answer 1

$\int \frac{1}{(x+1)(x+2)}dxlet,\frac{1}{(x+1)(x+2)}=\frac{A}{x+1}+\frac{B}{x+2}\Rightarrow 1=A(x+2)+B\left(\right(x+1)\Rightarrow 1=(A+B)x+(2A+B)$

comparing both side we get,

$A+B=0⟶\left(1\right)and2A+B=1⟶\left(2\right)$

solving (1) and (2) we get,

$A=1$ and $B=-1$

$\int \frac{1}{(x+1)(x+2)}dx=\int [\frac{1}{x+1}-\frac{1}{x+2}]dx=\log |x+1|-\log |x+2|+C=\log \left|\frac{x+1}{x+2}\right|+C$