The value of the integral `int_(0)^(pi//2) (sqrt""(cot x))/(sqrt""(cot x) + sqrt""(tan x))dx` is

Options

A. `(pi)/(4)`
B. `(pi)/(2)`
C. `pi`
D. none

Answer

To solve the integral

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

we can use a property of definite integrals. The property states that:

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

### Step 1: Apply the property to the integral

Let’s apply this property to our integral. We set \( a = \frac{\pi}{2} \), and we will find \( I \) in terms of \( I \) itself:

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

Using the property, we have:

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

### Step 2: Simplify the expressions

We know that:

\[ \cot\left(\frac{\pi}{2} - x\right) = \tan x \quad \text{and} \quad \tan\left(\frac{\pi}{2} - x\right) = \cot x. \]

Thus, we can rewrite the integral as:

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

### Step 3: Combine the two integrals

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

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

Let’s add these two equations:

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

### Step 4: Simplify the sum

Notice that the denominators are the same, so we can combine the fractions:

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

### Step 5: Evaluate the integral

The integral of 1 from 0 to \(\frac{\pi}{2}\) is simply:

\[ \int_{0}^{\frac{\pi}{2}} 1 \, dx = \frac{\pi}{2}. \]

Thus, we have:

\[ 2I = \frac{\pi}{2}. \]

### Step 6: Solve for \( I \)

Dividing both sides by 2 gives:

\[ I = \frac{\pi}{4}. \]

### Final Answer

The value of the integral is:

\[ \boxed{\frac{\pi}{4}}. \]