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Q. $ \int \sqrt{1+\cos \,x } \,dx$ is equal to

KEAMKEAM 2014Integrals

Solution:

Let $I=\int \sqrt{1+\cos \,x} \,d x=\int \sqrt{2 \cos ^{2} \frac{x}{2}} \,d x$
$=\int \sqrt{2} \,\cos \frac{x}{2}\, d x$
$=\frac{\sqrt{2} \sin \frac{x}{2}}{1 / 2}+C$
$=2 \sqrt{2} \,\sin \left(\frac{x}{2}\right)+C$