Evaluate _ - 4 ^ 4 ^2 x d x. - Vidyadip

Question

Evaluate $\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}} \sin ^2 x d x$.

✓

Answer

Here $\sin ^2 x$ is an even function.
So, $\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}} \sin ^2 x d x=2 \int_0^{\frac{\pi}{4}} \sin ^2 x d x$
$\left(\because \int_{-a}^a f(x) d x=2 \int_0^a f(x) d x\right)$
$=2 \int_0^{\frac{\pi}{4}} \frac{1-\cos 2 x}{2} d x$
$=2 \int_0^{\frac{\pi}{4}}(1-\cos 2 x) d x$
$=\left(x-\frac{1}{2} \sin 2 x\right)_0^{\frac{\pi}{4}}$
$=\left(\frac{\pi}{4}-\frac{1}{2} \sin \frac{\pi}{2}\right)-0=\frac{\pi}{4}-\frac{1}{2} \text { }$

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