Question

\( \int_{0}^{\pi / 2} \sqrt{1+\cos x} d x \)

Answer 1

Let I = \(\int ^ {\pi /2}_0\,\sqrt {1+cos\,x}\,dx\)

\(=\int ^ {\pi /2}_{0}\sqrt {2cos^ 2\,\frac {x}{2}dx}\)  \([\because\,cos 2A=2cos^2A-1]\)

\(= \sqrt {2} \int ^ {\pi/2} _ 0 \,\left(cos\frac{x}{2}\right)dx\)