Evaluate: ^ _ 0 x x . cosec x - Vidyadip

Question

Evaluate:
$\int\limits^{\pi}_{0}\frac{\text{x}\tan\text{x}}{\sec \text{x} . \text{cosec x}}$

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Answer

$\int\limits^{\pi}_{0}\frac{\text{x}\tan\text{x dx}}{\sec \text{x cosec x}}$
$\int\limits^{\pi}_{0}\text{x}\sin^{2}\text{x dx} $
$\text{Let I} = \int\limits^{\pi}_{0}\text{x}\sin^{2}\text{x dx}$
$= \int\limits^\pi_0(\pi -\text{x})\sin^{2}(\pi - \text{x) dx}$
$= \int\limits^{\pi}_{0}(\pi - \text{x)}\sin^{2}\text{x dx}$
$\text{2 I}= \pi\int\limits\sin^{2}\text{x dx} = \pi\int\limits^{\pi}_{0}\frac{1 - \cos\text{2 x}}{2}\text{dx}$
$= \frac{\pi}{2} \Bigg[\text{x} - \frac{\sin\text{2 x}}{2}\Bigg]^{\pi}_{0}$
$= \frac{\pi^{2}}{2}$
$\text{I} = \frac{\pi^{2}}{4}$

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