\(\int \limits_0^\frac \pi 2 \frac{x + \sin x}{1 + \cos x}dx\)
\(= \int \limits_0^\frac \pi 2 \frac{x }{1 + \cos x}dx + \int \limits_0^\frac \pi 2 \frac{ \sin x }{1 + \cos x}dx\)
\(= \int \limits_0^\frac \pi 2 \frac{x }{ 2\cos^2\frac x2}dx + \int \limits_0^\frac \pi 2 \frac{ 2\sin \frac x2 .\cos \frac x2}{ 2\cos^2\frac x2}dx\)
\(= \frac 12 \int\limits_0^{\frac \pi 2}x\sec^2\frac x2 dx + \int \limits_0^\frac \pi 2\tan \frac x2dx\)
\(= \frac 12 [x.2\tan \frac x2]_0^\frac\pi2 - \frac 12 \int \limits_0^\frac \pi 21.2\tan \frac x2 dx + \int \limits _0^\frac \pi 2 \tan \frac x2 dx\)
\(= \frac 12 \left[\frac \pi 2 . 2\tan \frac \pi 4 - 0\right]\)
\(= \frac \pi 2\)
