Question

If \[I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{1+\sqrt{\tan x}}\], what is the value of \[I\]?

Options

• \[\frac{\pi}{12}\]

• \[\frac{\pi}{6}\]

• \[\frac{\pi}{4}\]

• \[\frac{\pi}{3}\]

Answer 1

Adding the two forms of \[I\] gives \[2I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}dx=\frac{\pi}{6}\]. Hence \[I=\frac{\pi}{12}\].