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

Options

A. `(pi)/(2)`
B. `(pi)/(3)`
C. `(pi)/(4)`
D. `(pi)/(6)`

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \sqrt{\tan x}} \, dx \), we can use a symmetry property of definite integrals.

### Step-by-Step Solution:

1. **Set up the integral**:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \sqrt{\tan x}} \, dx \]

2. **Use the substitution**:
We can use the property of definite integrals:
\[ \int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a - x) \, dx \]
Here, let \( a = \frac{\pi}{2} \). Thus, we can write:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \sqrt{\tan\left(\frac{\pi}{2} - x\right)}} \, dx \]

3. **Simplify the expression**:
Using the identity \( \tan\left(\frac{\pi}{2} - x\right) = \cot x \), we have:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \sqrt{\cot x}} \, dx \]
Since \( \sqrt{\cot x} = \frac{1}{\sqrt{\tan x}} \), we can rewrite the integral:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \frac{1}{\sqrt{\tan x}}} \, dx \]

4. **Combine the two integrals**:
Now we have two expressions for \( I \):
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \sqrt{\tan x}} \, dx \]
and
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\tan x}}{\sqrt{\tan x} + 1} \, dx \]
Adding these two equations gives:
\[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{1}{1 + \sqrt{\tan x}} + \frac{\sqrt{\tan x}}{\sqrt{\tan x} + 1} \right) dx \]

5. **Simplify the integrand**:
The sum simplifies:
\[ \frac{1}{1 + \sqrt{\tan x}} + \frac{\sqrt{\tan x}}{\sqrt{\tan x} + 1} = \frac{1 + \tan x}{1 + \sqrt{\tan x}} = 1 \]
Thus, we have:
\[ 2I = \int_{0}^{\frac{\pi}{2}} 1 \, dx \]

6. **Evaluate the integral**:
\[ 2I = \left[ x \right]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2} \]

7. **Solve for \( I \)**:
\[ I = \frac{\pi}{4} \]

### Final Answer:
\[ I = \frac{\pi}{4} \]