Question

find the integral :

$\int\limits^{\frac{\pi}{2}}_{0} {\frac{1}{1+\sqrt{tan\ x}}} \, dx$

Thanks for a good solution. provide asap.
The answers is probably π, π/4, π/8, π/2, 2π ..like that.

Answer 1

$I = \int\limits^{\pi/2}_{0} {\frac{1}{\sqrt{tan\ x}+1}} \, dx\\let\ y=\pi/2-x,\ \ \ dx=-dy\\\\I= \int\limits^{\pi/2}_{0} {\frac{1}{1+\sqrt{cotx}}} \, dx = \int\limits^{\pi/2}_{0} {\frac{\sqrt{tan\ x}}{\sqrt{tan\ x}+1}} \, dx\\\\ Adding\ I+I= \int\limits^{\pi/2}_{0} {1} \, dx \\\\I=\pi/4$