` int _(0)^(pi//4) [ sqrt(tan x)+ sqrt(cot x)] ` dx is equal to

Answer

To solve the integral \( I = \int_0^{\frac{\pi}{4}} \left( \sqrt{\tan x} + \sqrt{\cot x} \right) \, dx \), we can follow these steps:

### Step 1: Rewrite the integral
We can express \(\tan x\) and \(\cot x\) in terms of sine and cosine:
\[ I = \int_0^{\frac{\pi}{4}} \left( \sqrt{\frac{\sin x}{\cos x}} + \sqrt{\frac{\cos x}{\sin x}} \right) \, dx \]

### Step 2: Combine the terms under a common denominator
We can combine the two square root terms:
\[ I = \int_0^{\frac{\pi}{4}} \left( \frac{\sqrt{\sin x}}{\sqrt{\cos x}} + \frac{\sqrt{\cos x}}{\sqrt{\sin x}} \right) \, dx \]
This can be rewritten as:
\[ I = \int_0^{\frac{\pi}{4}} \left( \frac{\sin x + \cos x}{\sqrt{\sin x \cos x}} \right) \, dx \]

### Step 3: Use the identity for \(\sin x + \cos x\)
We know that \(\sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right)\). Thus:
\[ I = \int_0^{\frac{\pi}{4}} \frac{\sqrt{2} \sin\left(x + \frac{\pi}{4}\right)}{\sqrt{\sin x \cos x}} \, dx \]

### Step 4: Change of variable
Let \( u = \frac{\pi}{4} - x \), then \( du = -dx \). The limits change as follows:
- When \( x = 0 \), \( u = \frac{\pi}{4} \)
- When \( x = \frac{\pi}{4} \), \( u = 0 \)

Thus, we can rewrite the integral:
\[ I = \int_{\frac{\pi}{4}}^0 \left( \sqrt{\tan\left(\frac{\pi}{4} - u\right)} + \sqrt{\cot\left(\frac{\pi}{4} - u\right)} \right)(-du) \]
This simplifies to:
\[ I = \int_0^{\frac{\pi}{4}} \left( \sqrt{\cot u} + \sqrt{\tan u} \right) \, du \]

### Step 5: Combine both integrals
Notice that:
\[ I = \int_0^{\frac{\pi}{4}} \left( \sqrt{\tan x} + \sqrt{\cot x} \right) \, dx = \int_0^{\frac{\pi}{4}} \left( \sqrt{\cot x} + \sqrt{\tan x} \right) \, dx \]
Thus, we can write:
\[ 2I = \int_0^{\frac{\pi}{4}} \left( \sqrt{\tan x} + \sqrt{\cot x} + \sqrt{\cot x} + \sqrt{\tan x} \right) \, dx = \int_0^{\frac{\pi}{4}} \left( \sqrt{\tan x} + \sqrt{\cot x} \right) \, dx \]
This means:
\[ 2I = \int_0^{\frac{\pi}{4}} 2 \, dx = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2} \]

### Step 6: Solve for \(I\)
Dividing both sides by 2:
\[ I = \frac{\pi}{4} \]

### Final Answer
Thus, the value of the integral is:
\[ \boxed{\frac{\pi}{4}} \]