Question

Find the integral of the function
$\frac{{\sin }^{2}x}{1+\cos x}$

Answer 1

We have,

$I=\int \frac{{\sin }^{2}x}{1+\cos x}dx$

We know that

$\cos x=1-2{\sin }^2\frac{x}{2}=2{\cos }^2\frac{x}{2}-1$

$\sin x=2\sin \frac{x}{2}\cos \frac{x}{2}$

Therefore,

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

$I=\int \frac{4{\sin }^2\frac{x}{2}{\cos }^2\frac{x}{2}}{2{\cos }^2\frac{x}{2}}dx$

$I=2\int {\sin }^2\frac{x}{2}dx$

$I=\int (1-\cos x)dx$

$I=x-\sin x+C$

Hence, this is the answer.