Question

Evaluate
${\int }_0^{\pi /2}\frac{x+\sin x}{1+\cos x}dx$

Answer 1

$\begin{array}{c}{\int }_0^{\frac{\pi }{2}}\frac{x+\sin x}{1+\cos x}dx\\ ={\int }_0^{\frac{\pi }{2}}\frac{x+2\sin \frac{x}{2}\cos \frac{x}{2}}{2{\cos }^2\frac{x}{2}}dx\\ ={\int }_0^{\frac{\pi }{2}}\left(x\frac{1}{2}{\sec }^2\frac{x}{2}+\tan \frac{x}{2}\right)dx\\ \therefore fromformula,\int \left[x{f}^{{}^{\prime }}\left(x\right)+f\left(x\right)\right]dx=xf\left(x\right)\\ ={\left[x\tan \frac{x}{2}\right]}_0^{\frac{\pi }{2}}\\ =\left(\frac{\pi }{2}\tan \frac{\pi }{4}\right)-0\\ =\frac{\pi }{2}\end{array}$