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

Options

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

Answer

To solve the integral

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

we can use a property of definite integrals. This 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 define \( I \) as follows:

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

Now, we will apply the property:

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

### Step 2: Rewrite the integral

Now we can rewrite this integral as:

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

### Step 3: Add both expressions for \( I \)

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

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

Adding these two equations gives:

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

### Step 4: Simplify the integrand

The integrand simplifies as follows:

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

Thus, we have:

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

### Step 5: Evaluate the integral

Now, we can evaluate the integral:

\[ 2I = \left[ x \right]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2}. \]

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

Dividing both sides by 2 gives:

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

### Final Answer

Therefore, the value of the integral is:

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