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

Question

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

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Answer

Let $\mathrm{I}=\int_0^{\pi / 2} \sqrt{1-\cos 4 x} \cdot d x$
$
\begin{aligned}
\mathrm{I}= & \int_0^{\pi / 2} \sqrt{2 \sin ^2 2 x} \cdot d x \\
& \left(\because 1-\cos \mathrm{A}=2 \sin ^2 \frac{\mathrm{A}}{2}\right) \\
= & \sqrt{2} \cdot \int_0^{\pi / 2} \sin 2 x \cdot d x \\
= & \sqrt{2} \cdot\left[\frac{-\cos 2 x}{2}\right]_0^{\pi / 2} \\
= & \frac{\sqrt{2}}{2} \cdot\left[\cos 2 \frac{\pi}{2}-\cos 0\right] \\
= & -\frac{\sqrt{2}}{2} \cdot[\cos \pi-\cos 0] \\
= & -\frac{\sqrt{2}}{2} \cdot(-1-1)=\sqrt{2} \\
\therefore \quad \int_0^{\pi / 2} & \sqrt{1-\cos 4 x} \cdot d x=\sqrt{2}
\end{aligned}
$

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