Question

Find its:-
$\int \frac{1}{x\left(x+1\right)}dx$

Answer 1

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

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

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

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

$=\log \left|x\right|-\log |x+1|+c$

$=\log \left|\frac{x}{x+1}\right|+c$