Evaluate: ^ _0x x sec x. cosec x dx. - Vidyadip

Question

Evaluate: $\int\limits^\pi_0\frac{x\tan x}{\text{sec }x.\text{ cosec }x}\text{ d}x.$

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Answer

$\text{Let I}=\int\limits^\pi_0\frac{\text{x tan x}}{\text{sec x cosec x}}\text{ dx}$
$\therefore\text{ I}=\int\limits^\pi_0\frac{(\pi-\text{x})\tan(\pi-\text{x})}{ \sec (\pi-\text{x}) \text{ cosec} (\pi-\text{x})}\text{ dx}$
$\Rightarrow\text{ I}=\int\limits^\pi_0\frac{(\pi-\text{x})\tan\text{x}}{ \sec\text{x}.\text{cosec}\text{ x}}\text{ dx}$
$\text{Adding we ge t, 2I}=\pi\int\limits^\pi_0\frac{\tan\text{x}}{ \sec\text{x}.\text{cosec}\text{ x}}\text{ dx}$
$\text{2I}=\pi\int\limits^\pi_0\sin^2\text{x}\text{ dx}$
$=2\pi\int\limits^{\pi/2}_0\frac{1-\cos2\text{x}}{2}\text{ dx}$
$=\pi\bigg[\Big(\text{x}-\frac{\sin2\text{x}}{2}\Big)\bigg]^{\pi/2}_0$
$=\pi.\frac{\pi}{2}=\frac{\pi^2}{2}\ \ \therefore\ \ \text{I}=\frac{\pi^2}{4}$

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