Solveeit Logo
Login

Question

Mathematics Question on integral

Evaluate:∫02πsin(π4+x2)dx\int\limits_{0}^{2\pi} sin\left(\frac{\pi}{4}+\frac{x}{2}\right)dx

A

−22-2\sqrt{2}

B

−2-2

C

2\sqrt{2}

D

222\sqrt{2}

Answer

222\sqrt{2}

Explanation

Solution

Let I=∫0π2sin(π4+x2)dxI = \int\limits_{0}^{\frac{\pi}{2}} sin \left(\frac{\pi}{4} + \frac{x}{2}\right) dx I=[−2 cos (π4+ x2)]02π=−2(cos (π4+π)−cos π4)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 π4+2 cos π4=2 2 = 2\,cos \,\frac{\pi}{4} + 2\,cos \,\frac{\pi}{4} = 2\,\sqrt{2}