Question

Solve:-
$\int \frac{\cos x}{1+\cos x}dx$

Answer 1

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

$=\int \text{cosec}x\cot xdx-\int {\cot }^{2}xdx=-\text{cosec}x-\int ({\text{cosec}}^{2}x-1)dx-\text{cosec}x-\int {\text{cosec}}^{2}xdx+\int dx=-\text{cosec}x+\cot x+x+C$