Question

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

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

Answer 1

The correct option is C $-\pi$

$I=\stackrel{2\pi }{\underset{0}{\int }}\left[\sin 2x\right(1+\cos 3x\left)\right]dx$
$I=\stackrel{\pi }{\underset{0}{\int }}\left(\right[\sin 2x+\sin 2x\cos 3x]+[-\sin 2x-\sin 2x\cos 3x\left]\right)dx$
$=\stackrel{\pi }{\underset{0}{\int }}-dx=-\pi$