The value of the integral $\int e^{\sin ^2 x}\left(\cos x+\cos ^3 x\right) \sin x d x$ is
Answer & explanation
Correct answer: option 2
Put $t=\sin ^2 x$
The integral reduces to
$I=\frac{1}{2} \int e^t(2-t) d t=\frac{3}{2} e^t-\frac{t e^t}{2}+c$
$=\frac{1}{2} e^{\sin ^2 x}\left(3-\sin ^2 x\right)+c$
$=e^{\sin ^2 x}\left(1+\frac{1}{2} \cos ^2 x\right)+c$
Hence (2) are the correct answer.