Evaluate the following integrals: ^ 2 _0 ^ 7 x ^ 7 x+ ^7x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{\tan^{7}\text{x}}{\tan^{7}\text{x}+\cot^7\text{x}}\text{ dx}$

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Answer

Let $\text{I}=\int\limits^{\frac{\pi}{2}}_0\frac{\tan^{7}\text{x}}{\tan^{7}\text{x}+\cot^7\text{x}}\text{ dx}\ ...(\text{i})$
Then,
$\text{I}=\int\limits^{\frac{\pi}{2}}_0\frac{\tan^{7\big(\frac{\pi}{2}-\text{x}\big)}}{\tan^{7}{\big(\frac{\pi}{2}-\text{x}\big)}+\cot^7{\big(\frac{\pi}{2}-\text{x}\big)}}\text{ dx}$ $\Bigg[\int\limits^{\text{a}}_0\text{f(x)}\text{dx}=\int\limits^{\text{a}}_0\text{f}(\text{a}-\text{x})\text{dx}\Bigg]$
$=\int\limits^{\frac{\pi}{2}}_0\frac{\cot^7\text{x}}{\cot^7\text{x}+\tan^{7}\text{x}}\text{ dx}\ ...(\text{ii})$
Adding (i) and (ii) we get
$2\text{I}=\int\limits^{\frac{\pi}{2}}_0\frac{\tan^{7}\text{x}+\cot^7\text{x}}{\tan^{7}\text{x}+\cot^7\text{x}}\text{ dx}$
$\Rightarrow2\text{I}=\int\limits^{\frac{\pi}{2}}_0\text{dx}$
$\Rightarrow2\text{I}=\Big[\text{x}\Big]^{\frac{\pi}{2}}_0$
$\Rightarrow2\text{I}=\frac{\pi}{2}-0=\frac{\pi}{2}$
$\Rightarrow\text{I}=\frac{\pi}{4}$

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