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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:
$\int\limits_{0}^{\frac{\pi}{2}}\frac{\sin^{\frac{3}{2}}\text{x}\ \text{dx}}{\sin^{\frac{3}{2}}\text{x}+\cos^{\frac{3}{2}}\text{x}}$
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Answer

$\text{Let}\ \text{I}=\int\limits_{0}^{\frac{\pi}{2}}\frac{\sin^{\frac{3}{2}}\text{x}\ }{\sin^{\frac{3}{2}}\text{x}+\cos^{\frac{3}{2}}\text{x}}\text{dx}$

$\Rightarrow\ \ \text{I}=\int\limits_{0}^{\frac{\pi}{2}}\frac{\sin^{\frac{3}{2}}\bigg(\frac{\pi}{2}-\text{x}\bigg)}{\sin^{\frac{3}{2}}\bigg(\frac{\pi}{2}-\text{x}\bigg)+\cos^{\frac{3}{2}}\bigg(\frac{\pi}{2}-\text{x}\bigg)}\text{dx}\ \ \bigg[\because\int\limits_{0}^{\text{a}}\text{f}\text{(x)}\ \text{dx}=\int\limits_{0}^{\text{a}}\text{f}(\text{a}-\text{x})\text{dx}=\bigg]$

$\Rightarrow\ \ \text{I}=\int\limits_{0}^{\frac{\pi}{2}}\frac{\cos^{\frac{3}{2}}\text{x}}{\cos^{\frac{3}{2}}\text{x}+\sin^{\frac{3}{2}}\text{x}}\text{dx}$

Adding eq. (i) and (ii),

$21=\int\limits_{0}^{\frac{\pi}{2}}\bigg[\frac{\sin^{\frac{3}{2}}\text{x}}{\sin^{\frac{3}{2}}\text{x}+\cos^{\frac{3}{2}}\text{x}}+\frac{\cos^{\frac{3}{2}}\text{x}}{\cos^{\frac{3}{2}}\text{x}+\sin^{\frac{3}{2}}\text{x}}\bigg]\text{dx}=\int\limits_{0}^{\frac{\pi}{2}}\bigg[\frac{\sin^{\frac{3}{2}}+\cos^{\frac{3}{2}}\text{x}}{\sin^{\frac{3}{2}}\text{x}+\cos^{\frac{3}{2}}\text{x}}\bigg]\text{dx}$

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

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