Question

$\int \frac{cosx}{1+cosx}dx$

Answer 1

We have,

$\int \frac{\cos x}{1+\cos x}dx$

On rationalize and we get,

$\int \frac{\cos x}{1+\cos x}dx$

$\Rightarrow \int \left(\frac{\cos x}{1+\cos x}\right)\left(\frac{1-\cos x}{1-\cos x}\right)dx$

$\Rightarrow \int \frac{\cos x(1-\cos x)}{1-{\cos }^{2}x}dx$

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

$\Rightarrow \int \frac{\cos x-{\cos }^{2}x}{{\sin }^{2}x}dx$

$\Rightarrow \int (\csc x\cot x-{\cot }^{2}x)dx$

$\Rightarrow \int \csc x\cot xdx-\int {\cot }^{2}xdx$

On integrating and we get,

$-\csc x+\cot x+C$

Hence, this is the answer.