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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals4 Marks
Question
By using the properties of definite integrals, evaluate the integral $\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$
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Answer

Given integral is: $\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$ 
Let $I=\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$ .....(i)
as $\left\{\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x\right\}$ 
$I=\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin \left(\frac{\pi}{2}-x\right)}}{\sqrt{\sin \left(\frac{\pi}{2}-x\right)}+\sqrt{\cos \left(\frac{\pi}{2}-x\right)}} d x$ 
$\Rightarrow \mathrm{I}=\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\cos \mathrm{x}}}{\sqrt{\cos \mathrm{x}}+\sqrt{\sin \mathrm{x}}} \mathrm{d} \mathrm{x}$ ......(ii)
Adding (i) and (ii), we get
$2 \mathrm{I}=\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin \mathrm{x}}+\sqrt{\cos \mathrm{x}}}{\sqrt{\sin \mathrm{x}}+\sqrt{\cos \mathrm{x}}} \mathrm{dx}$ 
$\Rightarrow 2 \mathrm{I}=\int_{0}^{\frac{\pi}{2}}[1] \mathrm{d} \mathrm{x}$ 
$\Rightarrow 2 \mathrm{I}=\frac{\pi}{2}-0$ 
$\Rightarrow 2 \mathrm{I}=\frac{\pi}{2}$
$\Rightarrow \mathrm{I}=\frac{\pi}{4}$

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