JEE Main 2026 — 21 January, Morning Shift — Question 25

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

$$\begin{aligned} & I=6 \int_{0}^{\pi}\left|4 \sin x \cos ^{2} x+2 \sin x \cos x\right| d x \end{aligned}$$

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

Put $\cos \mathrm{x}=\mathrm{t},-\sin \mathrm{xdx}=\mathrm{dt}$

$$\begin{aligned} & I=-12 \int_{1}^{-1}\left|2 t^{2}+t\right| d t & \end{aligned}$$

$I= 12\left(\int_{-1}^{-1 / 2}\left(2 t^{2}+t\right) d t+\int_{-1 / 2}^{0}-\left(2 t^{2}+t\right) d t+\int_{0}^{1}\left(2 t^{2}+t\right) d t\right)$

$I=17$