Question

Evaluate $\int \frac{{\cot }^{2}x}{(cose{c}^{2}x+cosecx)}dx$

Answer 1

$\int \frac{{\cot }^{2}xdx}{{\csc }^{2}x+\csc x}$

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

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

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

$=\int (1-\sin x)dx$

$=x+\cos x+c$