`int_(0)^(pi//2)(1)/(sqrt(tan x)-sqrt(cot x))dx=`

Options

A. `(pi)/(2)`
B. `(pi)/(4)`
C. 0
D. 1

Answer

To solve the integral

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{\sqrt{\tan x} - \sqrt{\cot x}} \, dx, \]

we can use the property of definite integrals that states:

\[ \int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a - x) \, dx. \]

### Step 1: Apply the property of definite integrals

We apply this property with \( a = \frac{\pi}{2} \):

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{\sqrt{\tan\left(\frac{\pi}{2} - x\right)} - \sqrt{\cot\left(\frac{\pi}{2} - x\right)}} \, dx. \]

### Step 2: Simplify the integrand

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:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{\sqrt{\cot x} - \sqrt{\tan x}} \, dx. \]

### Step 3: Rewrite the integral

Now we have two expressions for \( I \):

1. \( I = \int_{0}^{\frac{\pi}{2}} \frac{1}{\sqrt{\tan x} - \sqrt{\cot x}} \, dx \) (Equation 1)
2. \( I = \int_{0}^{\frac{\pi}{2}} \frac{1}{\sqrt{\cot x} - \sqrt{\tan x}} \, dx \) (Equation 2)

### Step 4: Add the two equations

Adding Equation 1 and Equation 2 gives:

\[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{1}{\sqrt{\tan x} - \sqrt{\cot x}} + \frac{1}{\sqrt{\cot x} - \sqrt{\tan x}} \right) \, dx. \]

### Step 5: Combine the fractions

Combining the fractions in the integral:

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

The numerator simplifies to zero:

\[ \sqrt{\cot x} - \sqrt{\tan x} + \sqrt{\tan x} - \sqrt{\cot x} = 0. \]

### Step 6: Conclusion

Thus, we have:

\[ 2I = \int_{0}^{\frac{\pi}{2}} 0 \, dx = 0. \]

This implies:

\[ I = 0. \]

### Final Answer

The value of the integral is:

\[ \int_{0}^{\frac{\pi}{2}} \frac{1}{\sqrt{\tan x} - \sqrt{\cot x}} \, dx = 0. \]