Question

Evaluate:$\int\limits_{0}^{2\pi} sin\left(\frac{\pi}{4}+\frac{x}{2}\right)dx $

• A. $-2\sqrt{2}$
• B. $-2$
• C. $\sqrt{2}$
• D. $2\sqrt{2}$

Answer 1

Answer: (D)

Let $I = \int\limits_{0}^{\frac{\pi}{2}} sin \left(\frac{\pi}{4} + \frac{x}{2}\right) dx$

$I= \left[-2\,cos\, \left(\frac{\pi}{4} +\,\frac{x}{2}\right)\right]_{0}^{2\pi} = -2 \left(cos\, \left(\frac{\pi}{4} + \pi\right) -cos\, \frac{\pi}{4}\right)$

$ = 2\,cos \,\frac{\pi}{4} + 2\,cos \,\frac{\pi}{4} = 2\,\sqrt{2}$