Ta có:
$\int\limits_0^{\frac{\pi}{2}}\dfrac{x+\sin x}{1+\cos x}dx$
$=\int\limits_0^{\frac{\pi}{2}}\dfrac{x+\sin x}{2\cos^2\dfrac{x}{2}}dx$
$=\int\limits_0^{\frac{\pi}{2}}\dfrac{x}{2\cos^2\dfrac{x}{2}}dx+\int\limits_0^{\frac{\pi}{2}}\dfrac{2\sin\dfrac{x}{2}\cos\dfrac{x}{2}}{2\cos^2\dfrac{x}{2}}dx$
$=\int\limits_0^{\frac{\pi}{2}}xd(\tan\dfrac{x}{2})+\int\limits_0^{\frac{\pi}{2}}\tan\dfrac{x}{2}dx$
$=x\tan\dfrac{x}{2} \left|\begin{array}{l}\dfrac{\pi}{2}\\0\end{array}\right.=\dfrac{\pi}{2}$