Evaluate: $\int_{0}^{\pi / 2} \frac{x+\sin x}{1+\cos x}\, dx$

CBSE Class 12 · 2011 · Definite IntegralsLong Answer

Evaluate:

\int_{0}^{π /2} \frac{x + s i n x}{1 + c o s x} d x

Answer: $\frac{π}{2}$

Use half-angle forms

1 + cos x = 2 cos^{2} \frac{x}{2}

and

sin x = 2 sin \frac{x}{2} cos \frac{x}{2}

\frac{x + sin x}{1 + cos x} = \frac{x}{2 cos ^{2} \frac{x}{2}} + tan \frac{x}{2} = \frac{x}{2} sec^{2} \frac{x}{2} + tan \frac{x}{2}

This is exactly

\frac{d}{d x} (x tan \frac{x}{2})

, so the integral telescopes:

I = [x tan \frac{x}{2}]_{0}^{π /2} = \frac{π}{2} tan \frac{π}{4} - 0 = \frac{π}{2}

CBSE Class 12 Board · 2011 · Subjective