Evaluate the definite integral _ 0 ^ 2 2 x d x - Vidyadip

Question

Evaluate the definite integral$\int_{0}^{\frac{\pi}{2}} \cos 2 x d x$

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Answer

Let $I=\int_{0}^{\frac{\pi}{2}} \cos 2 x d x$
$\Rightarrow \mathrm{I}=\left[\frac{\sin 2 \mathrm{x}}{2}\right]_{0}^{\pi / 2}$ [$\int \cos x d x=\sin x+c$ ]
$\Rightarrow I=\frac{1}{2}\left(\sin 2 \times \frac{\pi}{2}-\sin 2 \times 0\right)$
$\Rightarrow \mathrm{I}=\frac{1}{2}(\sin \pi-\sin 0)$
$\Rightarrow$ I = $\frac{1}{2}$(0 - 0) = 0
$\therefore \int_{0}^{\frac{\pi}{2}} \cos 2 x d x=0$

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