Evaluate of each of the following integral: `int_(pi//6)^(pi//3)1/(1+sqrt(tanx))dx`

Explanation

To evaluate the integral \( I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{1 + \sqrt{\tan x}} \, dx \), we will follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integrand using the identity \( \tan x = \frac{\sin x}{\cos x} \):
\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{1 + \sqrt{\frac{\sin x}{\cos x}}} \, dx \]
This can be rewritten as:
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