Question

The value of the definite integral $\underset{0}{\overset{\pi /2}{\int }}(\sin x+\cos x)\cdot \sqrt{\frac{e^x}{\sin x}}dx$ equals

• A. $2\sqrt{e^{\pi /2}}$
• B. $\sqrt{e^{\pi /2}}$
• C. $2\sqrt{e^{\pi /2}}\cdot \cos 1$
• D. $\frac{1}{2}{e}^{\pi /4}$

Answer 1

Answer: (A)

$I=\underset{0}{\overset{\pi /2}{\int }}\frac{e^x(\sin x+\cos x)}{\sqrt{e^x\sin x}}dx$
$\text{ put }{e}^x\sin x=t^2\Rightarrow e^x(\sin x+\cos x)dx=2tdt$
$\therefore {I=2\underset{0}{\overset{\pi /4}{\int }}dt=2t]}_0^{e^{\pi /4}}=2e^{\pi /4}=2\sqrt{e^{\pi /2}}$