If [x] stands for the greatest integer function, the value of `int_(4)^(10) ([x^(2)])/([x^(2)-28x +196]+[x^(2)])dx` is

Options

A. 0
B. 1
C. 3
D. None of these

Answer

To solve the integral

\[ \int_{4}^{10} \frac{[x^2]}{[x^2 - 28x + 196] + [x^2]} \, dx, \]

where \([x]\) denotes the greatest integer function, we will follow these steps:

### Step 1: Simplify the Expression Inside the Integral

First, we need to analyze the components of the integral. The expression \([x^2 - 28x + 196]\) can be rewritten as:

\[ [x^2 - 28x + 196] = [ (x - 14)^2 ]. \]

Since \((x - 14)^2\) is always non-negative, we can find the greatest integer value for different ranges of \(x\).

### Step 2: Evaluate the Greatest Integer Function

For \(x\) in the range from 4 to 10, we can compute \([x^2]\) and \([(x - 14)^2]\):

- When \(x = 4\), \(x^2 = 16\) and \([x^2] = 16\).
- When \(x = 10\), \(x^2 = 100\) and \([x^2] = 100\).

The values of \([x^2]\) will take integer values from 16 to 100 as \(x\) varies from 4 to 10.

### Step 3: Determine the Values of \([x^2 - 28x + 196]\)

Next, we calculate \([x^2 - 28x + 196]\):

\[ x^2 - 28x + 196 = (x - 14)^2. \]

- At \(x = 4\), \((4 - 14)^2 = 100\) and \([100] = 100\).
- At \(x = 10\), \((10 - 14)^2 = 16\) and \([16] = 16\).

Thus, as \(x\) varies from 4 to 10, \([(x - 14)^2]\) will also take integer values from 16 to 100.

### Step 4: Combine the Results

Now we can rewrite the integral:

\[ \int_{4}^{10} \frac{[x^2]}{[(x - 14)^2] + [x^2]} \, dx. \]

### Step 5: Use the Symmetry Property of the Integral

Using the property of definite integrals, we can express the integral as:

\[ \int_{4}^{10} f(x) \, dx = \int_{4}^{10} f(14 - x) \, dx. \]

This means we can evaluate the integral by substituting \(x\) with \(14 - x\) and adding the two integrals.

### Step 6: Calculate the Integral

Adding the two integrals gives us:

\[ 2I = \int_{4}^{10} \left( \frac{[x^2]}{[(x - 14)^2] + [x^2]} + \frac{[(14 - x)^2]}{[(14 - x - 14)^2] + [(14 - x)^2]} \right) \, dx. \]

The symmetry will simplify the calculation, and ultimately we find:

\[ I = 3. \]

### Final Answer

Thus, the value of the integral is:

\[ \boxed{3}. \]