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6 int 0 pi sin 3 x sin 2 x sin x dx is equal to | null (Mathematics) | DivineJEE

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Mathematics

JEE Mains 2026 21st January Shift 1

Mathematics

6 \int_{0}^{\pi}|(\sin 3 \mathrm{x}+\sin 2 \mathrm{x}+\sin \mathrm{x})| \mathrm{dx}

is equal to.......

Explanation

\quad 6 \int_{0}^{\pi}|2 \sin 2 \mathrm{x} \cos \mathrm{x}+\sin 2 \mathrm{x}| \mathrm{dx}\\

6 \int_{0}^{\pi}\left|4 \sin x \cos ^{2} x+2 \sin x \cos x\right| d x\\

I=12 \int_{0}^{\pi} \sin x\left|2 \cos ^{2} x+\cos x\right|\\

Put

\cos \mathrm{x}=\mathrm{t},-\sin \mathrm{x} \mathrm{dx}=\mathrm{dt}\\

\begin{aligned} & I=-12 \int_{1}^{-1}\left|2 t^{2}+t\right| d t \\ & I= 12\left(\int_{-1}^{-1 / 2}\left(2 t^2+t\right) d t+\int_{-1 / 2}^0-\left(2 t^2+t\right) d t+\int_0^1\left(2 t^2+t\right) d t\right) \end{aligned}

\\ \mathrm{I}=17

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