Question

Evaluate:

$\int \frac{\cot x}{1+{\sin }^{2}x}dx$

Answer 1

$\int \frac{cotx}{1+{sin}^{2}x}dx$

$\Rightarrow \int \frac{cosx}{sinx(1+{sin}^{2}x)}dx$

$sinx=t$

$cosdx=dt$

$\Rightarrow \int \frac{dt}{t(1+t^2)}=\int [\frac{-t}{1+t^2}+\frac{1}{t}]dt$

$\int \frac{dt}{t}-\int \frac{tdt}{1+t^2}$

$1+t^2=m$

$2tdt=dm$

$\Rightarrow Int-\frac{1}{2}\int \frac{dm}{m}$

$\Rightarrow Int-\frac{1}{2}Inm+c$

$=Int-\frac{1}{2}In(1+t^2)+c$