`int_(0)^(pi//2) (dx)/(sqrt(tan x)- sqrt(cot x))=`

Options

A. `pi//2`
B. `pi//4`
C. 0
D. none

Answer

To solve the integral

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{dx}{\sqrt{\tan x} - \sqrt{\cot x}}, \]

we will use a property of definite integrals. Let's proceed step by step.

### Step 1: Define the Integral
Let
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{dx}{\sqrt{\tan x} - \sqrt{\cot x}}. \]

### Step 2: Use the Property of Definite Integrals
We will use the property that states:
\[ \int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a - x) \, dx. \]
In our case, we will replace \(x\) with \(\frac{\pi}{2} - x\).

### Step 3: Substitute \(x\) with \(\frac{\pi}{2} - x\)
Now, we compute:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{dx}{\sqrt{\tan\left(\frac{\pi}{2} - x\right)} - \sqrt{\cot\left(\frac{\pi}{2} - x\right)}}. \]
Using the identities \(\tan\left(\frac{\pi}{2} - x\right) = \cot x\) and \(\cot\left(\frac{\pi}{2} - x\right) = \tan x\), we can rewrite the integral as:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{dx}{\sqrt{\cot x} - \sqrt{\tan x}}. \]

### Step 4: Simplify the Integral
Notice that:
\[ \sqrt{\cot x} - \sqrt{\tan x} = -(\sqrt{\tan x} - \sqrt{\cot x}). \]
Thus, we can express \(I\) as:
\[ I = -\int_{0}^{\frac{\pi}{2}} \frac{dx}{\sqrt{\tan x} - \sqrt{\cot x}} = -I. \]

### Step 5: Solve for \(I\)
From the equation \(I = -I\), we can add \(I\) to both sides:
\[ 2I = 0. \]
Thus, we find:
\[ I = 0. \]

### Conclusion
The value of the integral is:
\[ \int_{0}^{\frac{\pi}{2}} \frac{dx}{\sqrt{\tan x} - \sqrt{\cot x}} = 0. \]