Evaluate the following integrals: _ 0 ^ 2 1+ d - Vidyadip

Question

Evaluate the following integrals:$\int_{0}^\limits{\frac{\pi}{2}}\frac{\sin\theta}{\sqrt{1+\cos\theta}}\text{ d}\theta$

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Answer

Let $\text{I}=\int_{0}^\limits{\frac{\pi}{2}}\frac{\sin\theta}{\sqrt{1+\cos\theta}}\text{ d}\theta$ Let $\cos\theta=\text{t}$ Then, $-\sin\theta\text{ d}\theta=\text{dt}$ When $\theta=0,\text{t}=1$ and $\theta=\frac{\pi}{2},\text{t}=0$$\therefore\ \text{I}=\int_{0}^\limits{\frac{\pi}{2}}\frac{\sin\theta}{\sqrt{1+\cos\theta}}\text{ d}\theta$
$=\int_{1}^\limits{0}\frac{-\text{dt}}{\sqrt{1+\text{t}}}$
$=\int_{0}^\limits{1}\frac{\text{dt}}{\sqrt{1+\text{t}}}$
$=2\big[\sqrt{1+\text{t}}\big]^1_0$
$=2\big(\sqrt{2}-1\big)$

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