Question

Evaluate: ${\int }_0^{\frac{\pi }{2}}{e}^{{\sin }^{2}x}\sin 2xdx$

• A. $e$
• B. $e+1$
• C. $e-1$
• D. $2e+1$

Answer 1

Answer: (C)

$e-1$

${\int }_0^{\frac{\pi }{2}}{e}^{{\sin }^{2}x}\sin 2xdx$

Let $z={\sin }^{2}x$

$⟹dz=2\sin x\cos xdx=\sin 2xdx$

For $x=0,z=0$ and for $x=\frac{\pi }{2},z=1$.

Hence, integration becomes-

${\int }_0^1{e}^{z}dz$

$={\left[e^z\right]}_0^1$

$=e-1$