Question

The value of $\int^{2\pi}_{0} [\sin 2x ( 1+ \cos 3x)] dx$ , where [t] denotes the greatest integer function, is :

• A. $ - 2 \pi $
• B. $\pi$
• C. $ - \pi $
• D. $ 2 \pi $

Answer 1

Answer: (C)

$I = \int^{2\pi} \left[\sin2x\left(1+\cos3x\right)\right]dx$
$ I = \int^{\pi}_{0} \left(\left[\sin2x+\sin 2x \cos3x\right]+\left[-\sin2x -\sin2x \cos3x\right]\right)dx $
$ = \int^{\pi}_{0} -dx = - \pi $