Question

${\int }_0^{\pi }{e}^{{\sin }^{2}x}.{\cos }^{3}xdx$ is equal to

• A. $-1$
• B. $0$
• C. $1$
• D. $\pi$

Answer 1

The correct option is B $0$

Let $I={\int }_0^{\pi }{e}^{{\sin }^{2}x}.{\cos }^{3}xdx.....\left(i\right)$
$\Rightarrow I={\int }_0^{\pi }{e}^{{\sin }^2(\pi -x)}.{\cos }^3(\pi -x)dx$
$\Rightarrow I=-{\int }_0^{\pi }{e}^{{\sin }^{2}x}.{\cos }^{3}xdx.......\left(ii\right)$
On adding eqs. $\left(i\right)$ and $\left(ii\right)$, we get
$2I=0$
$\Rightarrow$ $I=0$ $\Rightarrow$ ${\int }_0^{\pi }{e}^{{\sin }^{2}x}.{\cos }^{3}xdx=0$