`int_(0)^(pi//2)(sqrt(tanx)+sqrt(cotx))\ dx`

Explanation

To solve the definite integral \( I = \int_{0}^{\frac{\pi}{2}} \left( \sqrt{\tan x} + \sqrt{\cot x} \right) \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We can express the integral in terms of sine and cosine:
\[ I = \int_{0}^{\frac{\pi}{2}} \left( \sqrt{\frac{\sin x}{\cos x}} + \sqrt{\frac{\cos x}{\sin x}} \right) \, dx \]
This simplifies to:
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