Question

Evaluate: $\int \frac{{\cos }^{3}x}{{\sin }^{2}x+\sin x}$

Answer 1

$I=\int \frac{{\cos }^{3}x}{{\sin }^{2}x+\sin x}dx$

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

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

Let $u=\sin x$

$du=\cos xdx$

$=\int \frac{(1-u^2)\cos x}{u^2+u}.\frac{du}{\cos x}$

$=\int \frac{(1+u)(1-u)}{u(u+1)}du$

$=\int \frac{1}{u}du-\int \frac{u}{u}du$

$=ln|u|-u+c$

$=ln|\sin x|-\sin x+c$.