Question

If the integrals of the function $\frac{{\sin }^{2}x}{1+\cos x}$ is $ax+b\sin x$.

Find $a+b$.

Answer 1

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

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

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

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

$=x-\sin x+C$

$a+b=1+(-1)=0$