Evaluate: _0^ /2 x + sin x 1 + cosx dx. - Vidyadip

Question

Evaluate:

$\int\limits_0^{\pi/2}\frac{\text{x + sin x}}{\text{1 + cosx}}\text{dx}$.

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Answer

$\text{I}=\int\limits_{0}^{\pi/2}\frac{\text{x}} {\text{1 + cos x}}\text{dx}+\int\limits_0^{\pi/2}\frac{\text{sin x}}{\text{1 + cos x}}\text{dx}$
$=\int\limits_0^{\pi/2}\cdot\text{x}\cdot\frac{1}{2}\sec^{2}\frac{\text{x}}{2}\text{dx}+\int\limits_0^{\pi/2}\tan\frac{\text{x}}{2}\text{dx}$
$=\Bigg[\text{x tan}\frac{\text{x}}{2}\Bigg]_{0}^{\pi/2}-\int\limits_0^{\pi/2}\tan\frac{\text{x}}{2}\text{dx}+\int\limits_0^{\pi/2}\tan\frac{\text{x}}{2}\text{dx}$
$=\frac{\pi}{2}1-0=\frac{\pi}{2}.$

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