Evaluate the following integrals: _ ^ 3 2 1- 2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int_{\pi}^\limits{\frac{3\pi}{2}}\sqrt{1-\cos2\text{x}}\text{ dx}$

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Answer

$\int_{\pi}^\limits{\frac{3\pi}{2}}\sqrt{1-\cos2\text{x}}\text{ dx}$
$=\int_{\pi}^\limits{\frac{3\pi}{2}}\sqrt{2\sin^2\text{x}}\text{ dx}$
$=\sqrt{2}\int_{\pi}^\limits{\frac{3\pi}{2}}\big[\sin\text{x}\big]\text{dx}$
$=\sqrt{2}\int_{\pi}^\limits{\frac{3\pi}{2}}\sin\text{x}\text{ dx}$ $(\sin\text{x}<0\text{ for }\pi\leq\text{x}\leq2\pi)$
$=\sqrt{2}\big[(-\cos\text{x})\big]^{\frac{3\pi}{2}}_\pi$
$=\sqrt{2}\Big(\cos\frac{3\pi}{2}-\cos\pi\Big)$
$=\sqrt{2}\big[0-(-1)\big]$
$=\sqrt{2}\times1$
$=\sqrt{2}$

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