If `int_(-2)^(5) f(x) dx= 7.5^(3)- 7(-2)^(3)` then f(x) is equal to

Options

A. `21x^(2)`
B. `3x^(2)`
C. `(3)/(2) x^(4)`
D. None of these

Answer

To solve the problem, we need to find the function \( f(x) \) such that

\[ \int_{-2}^{5} f(x) \, dx = 7 \cdot 5^3 - 7 \cdot (-2)^3. \]

### Step 1: Calculate the right-hand side

First, we will compute the right-hand side of the equation:

\[ 7 \cdot 5^3 - 7 \cdot (-2)^3. \]

Calculating \( 5^3 \) and \( (-2)^3 \):

\[ 5^3 = 125, \]
\[ (-2)^3 = -8. \]

Now substituting these values back into the equation:

\[ 7 \cdot 125 - 7 \cdot (-8) = 7 \cdot 125 + 7 \cdot 8. \]

Calculating \( 7 \cdot 125 \) and \( 7 \cdot 8 \):

\[ 7 \cdot 125 = 875, \]
\[ 7 \cdot 8 = 56. \]

Now add these two results:

\[ 875 + 56 = 931. \]

So, we have:

\[ \int_{-2}^{5} f(x) \, dx = 931. \]

### Step 2: Determine the form of \( f(x) \)

Since the integral is from \(-2\) to \(5\) and the right-hand side involves \(x^3\) terms, we can deduce that \( f(x) \) should be a polynomial of degree 2 (since integrating a polynomial of degree 2 gives a polynomial of degree 3).

Let’s assume:

\[ f(x) = ax^2. \]

### Step 3: Compute the integral

Now, we compute the integral:

\[ \int_{-2}^{5} ax^2 \, dx. \]

Using the formula for the integral of \( x^n \):

\[ \int x^n \, dx = \frac{x^{n+1}}{n+1}, \]

we have:

\[ \int ax^2 \, dx = a \cdot \frac{x^3}{3}. \]

Now, we evaluate this from \(-2\) to \(5\):

\[ \int_{-2}^{5} ax^2 \, dx = a \left[ \frac{5^3}{3} - \frac{(-2)^3}{3} \right]. \]

Calculating \( \frac{5^3}{3} \) and \( \frac{(-2)^3}{3} \):

\[ \frac{5^3}{3} = \frac{125}{3}, \]
\[ \frac{(-2)^3}{3} = \frac{-8}{3}. \]

Now substituting these values back:

\[ \int_{-2}^{5} ax^2 \, dx = a \left[ \frac{125}{3} - \frac{-8}{3} \right] = a \left[ \frac{125 + 8}{3} \right] = a \left[ \frac{133}{3} \right]. \]

### Step 4: Set the integral equal to 931

Now we set the integral equal to 931:

\[ a \cdot \frac{133}{3} = 931. \]

To solve for \( a \):

\[ a = 931 \cdot \frac{3}{133}. \]

Calculating \( a \):

\[ a = \frac{2793}{133} = 21. \]

### Conclusion

Thus, the function \( f(x) \) is:

\[ f(x) = 21x^2. \]