Evaluate _0^ ^2 x d x. - Vidyadip

Question

Evaluate $\int_0^\pi \sin ^2 x d x$.

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Answer

Let
$
\begin{aligned}
I & =\int_0^\pi \sin ^2 x d x=\int_0^\pi\left(\frac{1-\cos 2 x}{2}\right) d x \\
I & =\frac{1}{2}\left(x-\frac{\sin 2 x}{2}\right)_0^\pi \\
& =\frac{1}{2}\left(\left(\pi-\frac{1}{2} \sin 2 \pi\right)-(0-0)\right) \\
& =\frac{1}{2}(\pi-0)=\frac{\pi}{2} \text { }
\end{aligned}
$

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