x + sin x 1 + cos x ? d x is equal to :

Question

$\int \frac{x + \sin x}{1 + \cos x} d x$ is equal to :

Smart Achievers · Accepted Answer

(4)

Ans. $x \cdot \tan \frac{x}{2} + C$

Explanation :

$I = \int \frac{x + \sin x}{1 + \cos x} d x$

$= \int \frac{x}{1 + \cos x} d x + \int \frac{\sin x}{1 + \cos x} d x$

$= \int \frac{x}{2 \cos^{2} \frac{x}{2}} d x + \int \frac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \cos^{2} \frac{x}{2}} d x$

$= \frac{1}{2} \int x \sec^{2} \frac{x}{2} d x + \int \tan \frac{x}{2} d x$

$= \frac{1}{2} [x \cdot \tan \frac{x}{2} \cdot 2 - \int \tan \frac{x}{2} \cdot 2 d x] + \int \tan \frac{x}{2} d x$

$= x \cdot \tan \frac{x}{2} + C$ $[By cancelling the last two integrals]$

Want Us to Email you About Special Offers & Updates?