Evaluate: _0^ 2 1- 4 x d x - Vidyadip

Question

Evaluate: $\int_0^{\frac{\pi}{2}} \sqrt{1-\cos 4 x} d x$

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Answer

$\int_0^{\frac{\pi}{2}} \sqrt{1-\cos 4 x} d x$
$=\int_0^{\frac{\pi}{2}} \sqrt{2 \sin ^2 2 x} d x \quad \ldots \ldots . .\left[1-\cos \theta=2 \sin ^2 \frac{\theta}{2}\right]$
$=\sqrt{2} \int_0^{\frac{\pi}{2}} \sin 2 x d x$
$=\sqrt{2}\left[\frac{-\cos 2 x}{2}\right]_0^{\frac{\pi}{2}}$
$=\frac{\sqrt{2}}{2}\left[\cos 2 \frac{\pi}{2}-\cos 0\right]$
$=-\frac{\sqrt{2}}{2}[\cos \pi-\cos 0]$
$=-\frac{\sqrt{2}}{2}(-1-1)$
$=\sqrt{2}$

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