QJEEM19Ma2d76: JEE Main 2019 Mathematics - Class 12 JEEM19Ma2d76 Solution: The value of ∫ 0 2 π [ sin ⁡ 2 x ( 1 + cos ⁡ 3 x ) ] d x ,…

Question

Verified Solution • 2025–26 Session moderate

The value of

\int_{0}^{2 π} [\sin 2 x (1 + \cos 3 x)] d x

,
where [t] denotes the greatest integer function is :

Accepted Answer

I = \int_{0}^{2 π} [\sin 2 x (1 + \cos 3 x)] d x

.... (i)

∴ \int_{0}^{a} f (x) = \int_{0}^{a} f (a - x) d x

∵ I = \int_{0}^{2 π} [- \sin 2 x (1 + \cos 3 x)] d x

By (i) + (ii)

\Rightarrow 2 I = \int_{0}^{2 π} (- 1) d x

\Rightarrow 2 I = - (x)_{0}^{2 π}

\Rightarrow

I = –

π

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