• # question_answer The value of $\int\limits_{0}^{2\pi }{\left[ \sin 2x(1+cos3x) \right]}dx,$where [t] denotes the greatest integer function, is : [JEE Main 10-4-2019 Morning] A) $-2\pi$                                   B) $\pi$ C) $-\pi$ D) $2\pi$

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