Evaluate `int_(pi//6)^(pi//3) (d x )/(1+sqrt(tan x))`

Answer

To evaluate the integral

\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \sqrt{\tan x}}, \]

we can use the property of definite integrals that states:

\[ \int_a^b f(x) \, dx = \int_a^b f(a + b - x) \, dx. \]

### Step 1: Identify the limits and the function
Here, \( a = \frac{\pi}{6} \) and \( b = \frac{\pi}{3} \). Thus, \( a + b = \frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{6} + \frac{2\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \).

### Step 2: Substitute into the integral
Using the property mentioned, we can rewrite the integral:

\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \sqrt{\tan x}} = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \sqrt{\tan\left(\frac{\pi}{2} - x\right)}}. \]

### Step 3: Simplify the function
Using the identity \( \tan\left(\frac{\pi}{2} - x\right) = \cot x \), we have:

\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \sqrt{\cot x}}. \]

### Step 4: Rewrite \(\sqrt{\cot x}\)
We know that \( \cot x = \frac{1}{\tan x} \), so:

\[ \sqrt{\cot x} = \frac{1}{\sqrt{\tan x}}. \]

Thus, we can rewrite the integral:

\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \frac{1}{\sqrt{\tan x}}} = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\tan x}}{\sqrt{\tan x} + 1} \, dx. \]

### Step 5: Combine the two integrals
Now we have two expressions for \( I \):

1. \( I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \sqrt{\tan x}} \)
2. \( I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\tan x}}{\sqrt{\tan x} + 1} \, dx \)

Adding these two expressions gives:

\[ 2I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \left( \frac{1}{1 + \sqrt{\tan x}} + \frac{\sqrt{\tan x}}{\sqrt{\tan x} + 1} \right) dx. \]

### Step 6: Simplify the combined integral
The combined expression simplifies to:

\[ \frac{1 + \sqrt{\tan x}}{1 + \sqrt{\tan x}} = 1. \]

Thus, we have:

\[ 2I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} 1 \, dx = \left[ x \right]_{\frac{\pi}{6}}^{\frac{\pi}{3}} = \frac{\pi}{3} - \frac{\pi}{6} = \frac{\pi}{6}. \]

### Step 7: Solve for \( I \)
Dividing both sides by 2 gives:

\[ I = \frac{\pi}{6} \cdot \frac{1}{2} = \frac{\pi}{12}. \]

### Final Answer
Thus, the value of the integral is:

\[ \boxed{\frac{\pi}{12}}. \]