Question

Evaluate :

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

• A. $\log |\sin x|-\sin x+C$
• B. $\log |\cos x|-\cos x+C$
• C. $\log |\sin x|-\cos x+C$
• D. None of these

Answer 1

Answer: (A)

$\log |\sin x|-\sin x+C$

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

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

Let $\sin x=t$. Then, $d\left(\sin x\right)=dt$ or, $\cos xdx=dt$.

Therefore,

$I=\int \frac{1-t^2}{t(1+t)}dt=\int \frac{1-t}{t}dt=\int (\frac{1}{t}-1)dt$

$I=\log |t|-t+C$

$I=\log |\sin x|-\sin x+C$