\(\int \frac{\cos ^3(x)}{\sin ^2(x)+\sin (x)} d x=\)
\(\int \frac{\cos ^3(x)}{\sin ^2(x)+\sin (x)} d x=\)
\(\log |\sin (x)|+\sin (x)+c\)
\(\log |\sin (x)|+\cos (x)+c\)
\(\log |\cos (x)|-\sin (x)+c\)
\(\log |\sin (x)|-\sin (x)+c\)
Solution
\(\begin{aligned}
I & =\int \frac{\cos ^3 x}{\sin ^2 x+\sin x} \cdot d x \\
& =\int \frac{\left(1-\sin ^2 x\right) \cos x}{\sin ^2 x+\sin x} \cdot d x \\
\sin x & =t \Rightarrow \cos x d x=d t \\
\therefore \quad I & =\int \frac{1-t^2}{t^2+t} \cdot d t=\int \frac{(1-t)(1+t)}{t(1+t)} d t \\
& =\int\left(\frac{1}{t}-1\right) d t=\log t-t+C \\
& =\log |\sin x|-\sin x+C
\end{aligned}\)