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INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS3 Marks
Question
By using the properties of definite integral, evaluate the integral in Exercise:
$\int^{\frac{\pi}{2}}_{0}\frac{\sqrt{\sin\text{x}}}{\sqrt{\sin\text{x}}+\sqrt{\cos\text{x}}}\text{dx}$
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Answer

$\text{Let}\ \text{I}=\int\limits_{0}^{\frac{\pi}{2}}\frac{\sqrt{\sin\text{x}}}{\sqrt{\sin\text{x}}+\sqrt{\cos\text{x}}}\text{dx}$

$\Rightarrow\ \ \ \text{I}=\int\limits_{0}^{\frac{\pi}{2}}\frac{\sqrt{\sin\bigg(\frac{\pi}{2}-\text{x}}\bigg)}{\sqrt{\sin\bigg(\frac{\pi}{2}-\text{x}}\bigg)+\sqrt{\cos\bigg(\frac{\pi}{2}-\text{x}\bigg)}}\text{dx}\ \ \bigg[\because\int\limits_{0}^{\text{a}}\text{f}\text{(x)}\ \text{dx}=\int\limits_{0}^{\text{a}}\text{f}(\text{a}-\text{x})\text{dx}=\bigg]$

$\Rightarrow\ \ \text{I}=\int\limits_{0}^{\frac{\pi}{2}}\frac{\sqrt{\cos\text{x}}}{\sqrt{\cos\text{x}+\sqrt{\sin\text{x}}}}\text{dx}$

Adding eq. (i) and (ii),

$21=\int\limits_{0}^{\frac{\pi}{2}}\bigg(\frac{\sqrt{\sin\text{x}}}{\sqrt{\sin\text{x}+\sqrt{\cos\text{x}}}}+\frac{\sqrt{\cos\text{x}}}{\sqrt{\cos\text{x}+\sqrt{\sin\text{x}}}}\bigg)\text{dx}=\int\limits_{0}^{\frac{\pi}{2}}\bigg(\frac{\sqrt{\sin\text{x}}+\sqrt{\cos\text{x}}}{\sqrt{\sin\text{x}}+\sqrt{\cos\text{x}}}\bigg)\text{dx}$

$\Rightarrow\ \ \ \ 21=\int\limits_{0}^{\frac{\pi}{2}}1\ \text{dx}=\bigg(\text{x}^{\frac{\pi}{2}}_{0}\bigg)\ \ \ \Rightarrow\ \ \ \ 21=\frac{\pi}{2}\ \ \Rightarrow\ \text{I}=\frac{\pi}{4}$

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