Question

Evaluate : $\int \frac{x^2-1}{\left(x+1\right)\left(x+2\right)}dx$

Answer 1

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

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

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

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

$=$ $\int \frac{(x+2-3)}{(x+2)}dx$

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

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

$=$ $\int 1dx-$ $\int \frac{3}{(x+2)}dx$

$=$ $\int 1dx-3$ $\int \frac{1}{(x+2)}dx$

$=$ $x-3$ $\ln (x+2)$ $+C$