The value of the integral `int_(0)^(pi//2)(f(x))/(f(x)+f(pi/(2)-x))dx` is

Options

A. `pi//4`
B. `pi//2`
C. `pi`
D. 0

Answer

To solve the integral

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

we will use a property of definite integrals.

### Step 1: Define the Integral
Let

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

### Step 2: Use the Property of Definite Integrals
We can use the property of definite integrals which states that

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

In our case, we set \( a = 0 \) and \( b = \frac{\pi}{2} \). Thus, we have:

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

### Step 3: Change of Variable
Now, we can change the variable in the integral. Let \( u = \frac{\pi}{2} - x \). Then \( du = -dx \), and the limits change as follows:
- When \( x = 0 \), \( u = \frac{\pi}{2} \).
- When \( x = \frac{\pi}{2} \), \( u = 0 \).

Thus, we rewrite the integral:

\[ I = \int_{\frac{\pi}{2}}^{0} \frac{f(u)}{f(u) + f\left(\frac{\pi}{2} - u\right)} (-du) = \int_{0}^{\frac{\pi}{2}} \frac{f(u)}{f(u) + f\left(\frac{\pi}{2} - u\right)} \, du. \]

### Step 4: Combine the Two Integrals
Now we have two expressions for \( I \):

1. \( I = \int_{0}^{\frac{\pi}{2}} \frac{f(x)}{f(x) + f\left(\frac{\pi}{2} - x\right)} \, dx \) (Equation 1)
2. \( I = \int_{0}^{\frac{\pi}{2}} \frac{f\left(\frac{\pi}{2} - x\right)}{f\left(\frac{\pi}{2} - x\right) + f(x)} \, dx \) (Equation 2)

### Step 5: Add the Two Equations
Now, we add Equation 1 and Equation 2:

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

### Step 6: Simplify the Expression
The sum inside the integral simplifies to:

\[ \frac{f(x) + f\left(\frac{\pi}{2} - x\right)}{f(x) + f\left(\frac{\pi}{2} - x\right)} = 1. \]

Thus, we have:

\[ 2I = \int_{0}^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2}. \]

### Step 7: Solve for \( I \)
Now, dividing both sides by 2 gives:

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

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

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