Question

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

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

Answer 1

Answer: (D)

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

$I={\left[-2cos\left(\frac{\pi }{4}+\frac{x}{2}\right)\right]}_0^{2\pi }=-2\left(cos\left(\frac{\pi }{4}+\pi \right)-cos\frac{\pi }{4}\right)$

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