If the function `f(x)` is symmetric about the line `x=3`, then the value of the integral `I=int_(-2)^(8)(f(x))/(f(x)+f(6-x))dx` is

Options

A. 0
B. 5
C. 10
D. 16

Answer

To solve the integral \( I = \int_{-2}^{8} \frac{f(x)}{f(x) + f(6 - x)} \, dx \), given that the function \( f(x) \) is symmetric about the line \( x = 3 \), we can follow these steps:

### Step 1: Use the property of symmetry
Since \( f(x) \) is symmetric about \( x = 3 \), we have:
\[ f(6 - x) = f(x) \]
This property will be useful in simplifying the integral.

### Step 2: Rewrite the integral
We can rewrite the integral \( I \) using the property of symmetry:
\[ I = \int_{-2}^{8} \frac{f(x)}{f(x) + f(6 - x)} \, dx = \int_{-2}^{8} \frac{f(x)}{f(x) + f(x)} \, dx = \int_{-2}^{8} \frac{f(x)}{2f(x)} \, dx \]
This simplifies to:
\[ I = \int_{-2}^{8} \frac{1}{2} \, dx \]

### Step 3: Evaluate the integral
Now we can evaluate the integral:
\[ I = \frac{1}{2} \int_{-2}^{8} 1 \, dx = \frac{1}{2} [x]_{-2}^{8} = \frac{1}{2} [8 - (-2)] = \frac{1}{2} [8 + 2] = \frac{1}{2} [10] = 5 \]

### Final Result
Thus, the value of the integral \( I \) is:
\[ \boxed{5} \]