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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:
$\int^{\frac{\pi}{2}}_{0}\frac{\cos^{5}\text{x}\ \text{dx}}{\sin^{5}\text{x}+\cos^{5}\text{x}}$
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Answer

$\text{Let}\text{I}=\int\limits_{0}^{\frac{\pi}{2}}\frac{\cos^{5}\text{x}\ }{\sin^{5}\text{x}+\cos^{5}\text{x}}\text{dx}$

$\Rightarrow\ \text{I}=\int\limits_{0}^{\frac{\pi}{2}}\frac{\cos^{5}\bigg(\frac{\pi}{2}-\text{x}\bigg)}{\sin^{5}\bigg(\frac{\pi}{2}-\text{x}\bigg)+\cos^{5}\bigg(\frac{\pi}{2}-\text{x}\bigg)}\text{dx}\ \ \bigg[\because\int\limits_{0}^{\text{a}}\text{f}\text{(x)}\ \text{dx}=\int\limits_{0}^{\text{a}}\text{f}(\text{(a}-\text{x)}\text{dx}=\bigg]$

$\Rightarrow\ \ \text{I}=\int\limits_{0}^{\frac{\pi}{2}}\frac{\sin^{5}\text{x}}{\cos^{5}\text{x}+\sin^{5}\text{x}}\text{dx}$

Adding eq. (i) and (ii),

$21=\int\limits_{0}^{\frac{\pi}{2}}\bigg(\frac{\cos^{5}\text{x}}{\sin^{5}\text{x}+\cos^{5}\text{x}}+\frac{\sin^{5}\text{x}}{\cos^{5}\text{x}+\sin^{5}\text{x}}\bigg)\text{dx}=\int\limits_{0}^{\frac{\pi}{2}}\bigg(\frac{\cos^{5}\text{x}+\sin^{5}\text{x}}{\sin^{5}\text{x}+\cos{5}\text{x}}\bigg)\text{dx}$

$\Rightarrow\ \ 21=\int\limits_{0}^{\frac{\pi}{2}}1\ \text{dx}=\bigg(\text{x}^{\frac{\pi}{2}}_{0}\bigg)\ \Rightarrow21=\frac{\pi}{2}\ \Rightarrow\text{I}=\frac{\pi}{4}$

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