Question:
The value of $\int_{0}^{2 \pi}[\sin 2 x(1+\cos 3 x)] d x$, where [t] denotes the greatest integer function, is:
Correct Option: , 2
Solution:
$I=\int_{0}^{2 \pi}[\sin 2 x(1+\cos 3 x)] d x$.....(1)
$\because \int_{0}^{a} f(x)=\int_{0}^{a} f(a-x) d x$
$\therefore I=\int_{0}^{2 \pi}[-\sin 2 x(1+\cos 3 x)] d x$....(2)
From (1) + (2), we get;
$2 I=\int_{0}^{2 \pi}(-1) d x \Rightarrow 2 I=-(x)_{0}^{2 \pi} \Rightarrow I=-\pi$
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