Question

Evaluate: ${\int }_0^{2\pi }{e}^{x/2}\sin (\frac{1}{2}x+\frac{\pi }{4})dx$

• A. $2\sqrt{2}$
• B. $0$
• C. $2\pi$
• D. $e^{\pi }$

Answer 1

The correct option is B $0$

Let

$I={\int }_0^{2\pi }{e}^{x/2}\sin (\frac{x}{2}+\frac{\pi }{4})dx$

Let $\frac{x}{2}=t$

$\therefore I={\int }_0^{\pi }{e}^t\left(\sin (t+\frac{\pi }{4})\right)\cdot 2\cdot dt$

$=2{\int }_0^{\pi }{e}^t\left(\frac{1}{\sqrt{2}}\right(\sin t+\cos t\left)\right)dt$

$=\sqrt{2}{\int }_0^{\pi }{e}^t(\sin t+\cos t)dt(\therefore \frac{d}{dt}(e^t\sin t)=e^t(\sin t+\cos t\left)\right)$

$=\sqrt{2}[e^t\sin t{]}_0^{\pi }$

$=0$.