Evaluate: `int_(3)^(8) ((11-x)^(2))/(x^(2)+(11-x)^(2))dx`

Answer

To evaluate the integral

\[ I = \int_{3}^{8} \frac{(11-x)^{2}}{x^{2} + (11-x)^{2}} \, dx, \]

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

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

In our case, \( a = 3 \) and \( b = 8 \), so \( a + b = 11 \). We will evaluate \( I \) using this property.

### Step 1: Apply the property
We can rewrite the integral as follows:

\[ I = \int_{3}^{8} \frac{(11 - (11 - x))^{2}}{(11 - x)^{2} + (11 - (11 - x))^{2}} \, dx. \]

This simplifies to:

\[ I = \int_{3}^{8} \frac{x^{2}}{(11 - x)^{2} + x^{2}} \, dx. \]

### Step 2: Add the two integrals
Now we have two expressions for \( I \):

1. \( I = \int_{3}^{8} \frac{(11-x)^{2}}{x^{2} + (11-x)^{2}} \, dx \)
2. \( I = \int_{3}^{8} \frac{x^{2}}{(11-x)^{2} + x^{2}} \, dx \)

Adding these two integrals gives:

\[ 2I = \int_{3}^{8} \left( \frac{(11-x)^{2}}{x^{2} + (11-x)^{2}} + \frac{x^{2}}{(11-x)^{2} + x^{2}} \right) \, dx. \]

### Step 3: Simplify the integrand
The sum of the fractions in the integrand simplifies as follows:

\[ \frac{(11-x)^{2} + x^{2}}{x^{2} + (11-x)^{2}} = \frac{(11-x)^{2} + x^{2}}{(11-x)^{2} + x^{2}} = 1. \]

Thus, we have:

\[ 2I = \int_{3}^{8} 1 \, dx. \]

### Step 4: Evaluate the integral
Now we can evaluate the integral:

\[ 2I = \int_{3}^{8} 1 \, dx = [x]_{3}^{8} = 8 - 3 = 5. \]

### Step 5: Solve for \( I \)
Now, we can solve for \( I \):

\[ 2I = 5 \implies I = \frac{5}{2}. \]

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

\[ \boxed{\frac{5}{2}}. \]