Question

The value of ${\int }_0^{\pi }{e}^{{\sin }^{2}x}{\cos }^{3}xdx$ is

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

Answer 1

The correct option is B $0$

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

Hence, ${\int }_0^{\pi }{e}^{{\sin }^{2}x}{\cos }^{3}xdx=0$