Question

Evaluate $\int \frac{x}{(x+1)(x+2)}dx$.

Answer 1

$\frac{x}{(x+1)(x+2)}=\frac{A}{x+1}+\frac{B}{x+2}$

$x=A(x+2)+B(x+1)$

On comparing, $A=-1,B=2$

Hence, $\frac{x}{(x+1)(x+2)}=\frac{-1}{(x+1)}+\frac{2}{(x+2)}$

Therefore,

$I=\int \frac{x}{(x+1)(x+2)}dx=-1\int \frac{dx}{x+1}+2\int \frac{dx}{x+2}$

$I=-\log |x+1|+2\log |x+2|+C$