Evaluate
`int_(4)^(6)([x^(2)])/([x^(2)-20x+100]+[x^(2)])dx`
where [.] denotes the greatest integer function.

Answer

To evaluate the integral

\[ I = \int_{4}^{6} \frac{[x^2]}{[x^2 - 20x + 100] + [x^2]} \, dx \]

where \([.]\) denotes the greatest integer function, we will follow a systematic approach.

### Step 1: Simplify the expression inside the integral

First, we recognize that:

\[ x^2 - 20x + 100 = (x - 10)^2 \]

Thus, we can rewrite the integral as:

\[ I = \int_{4}^{6} \frac{[x^2]}{[(x - 10)^2] + [x^2]} \, dx \]

### Step 2: Identify the greatest integer values

Next, we need to evaluate the values of \([x^2]\) and \([(x - 10)^2]\) for \(x\) in the interval \([4, 6]\).

- For \(x = 4\), \(x^2 = 16\) and \([x^2] = 16\).
- For \(x = 5\), \(x^2 = 25\) and \([x^2] = 25\).
- For \(x = 6\), \(x^2 = 36\) and \([x^2] = 36\).

Now, we calculate \((x - 10)^2\):

- For \(x = 4\), \((4 - 10)^2 = 36\) and \([(x - 10)^2] = 36\).
- For \(x = 5\), \((5 - 10)^2 = 25\) and \([(x - 10)^2] = 25\).
- For \(x = 6\), \((6 - 10)^2 = 16\) and \([(x - 10)^2] = 16\).

### Step 3: Evaluate the integral using symmetry

Using the property of definite integrals, we can apply the transformation \(x \to 10 - x\):

\[ I = \int_{4}^{6} \frac{[x^2]}{[(x - 10)^2] + [x^2]} \, dx = \int_{4}^{6} \frac{[10 - x]^2}{[x^2]} \, dx \]

This means we can express the integral as:

\[ I = \int_{4}^{6} \frac{[10 - x]^2}{[(10 - x - 10)^2] + [10 - x]^2} \, dx \]

### Step 4: Combine the two integrals

Adding the two forms of the integral gives:

\[ 2I = \int_{4}^{6} \left( \frac{[x^2]}{[(x - 10)^2] + [x^2]} + \frac{[10 - x]^2}{[(10 - x - 10)^2] + [10 - x]^2} \right) \, dx \]

The denominators are the same, and thus we can simplify:

\[ 2I = \int_{4}^{6} 1 \, dx \]

### Step 5: Evaluate the integral

Now we compute:

\[ 2I = \int_{4}^{6} 1 \, dx = [x]_{4}^{6} = 6 - 4 = 2 \]

Thus, we have:

\[ I = \frac{2}{2} = 1 \]

### Final Answer

The value of the integral is

\[ \boxed{1} \]