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To find the average value of the function \( f(x) = x^3 + 5x - 2 \) over the interval \([2, 5]\), we use the formula for the average value of a function over an interval \([a, b]\):

\[
\text{Average Value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx
\]

Here, \( f(x) = x^3 + 5x - 2 \), \( a = 2 \), and \( b = 5 \). Let's compute this step by step.

### Step 1: Find the integral of \( f(x) \)
We need to compute the indefinite integral of \( f(x) = x^3 + 5x - 2 \):

\[
\int f(x) \, dx = \int (x^3 + 5x - 2) \, dx = \frac{x^4}{4} + \frac{5x^2}{2} - 2x + C
\]

### Step 2: Evaluate the definite integral
Now we evaluate the definite integral from 2 to 5:

\[
\int_2^5 (x^3 + 5x - 2) \, dx = \left[ \frac{x^4}{4} + \frac{5x^2}{2} - 2x \right]_2^5
\]

First, calculate at \( x = 5 \):

\[
\frac{5^4}{4} + \frac{5 \times 5^2}{2} - 2 \times 5 = \frac{625}{4} + \frac{125}{2} - 10 = 156.25 + 62.5 - 10 = 208.75
\]

Next, calculate at \( x = 2 \):

\[
\frac{2^4}{4} + \frac{5 \times 2^2}{2} - 2 \times 2 = \frac{16}{4} + \frac{20}{2} - 4 = 4 + 10 - 4 = 10
\]

Now subtract:

\[
208.75 - 10 = 198.75
\]

### Step 3: Compute the average value
Now divide by the length of the interval \( b - a = 5 - 2 = 3 \):

\[
\text{Average Value} = \frac{1}{3} \times 198.75 = 66.25
\]

Thus, the average value of the function over the interval \([2, 5]\) is:

\[
\boxed{66.25}
\]

Would you like any further details or have any other questions?

Here are five related questions to explore:
1. How does the average value formula apply to functions that are negative on certain intervals?
2. What is the geometric interpretation of the average value of a function over an interval?
3. Can you find the average value of \( f(x) = x^3 + 5x - 2 \) over a different interval, say \([0, 4]\)?
4. How does the presence of a cubic term in the function affect the shape of the graph?
5. What role does definite integration play in finding the area under curves?

**Tip:** When working with definite integrals, always carefully evaluate the function at the upper and lower limits to avoid errors.

Find the average value of the function f(x) = x^3 + 5x - 2 over the interval [2, 5].

Learn how to find the average value of the function f(x) = x^3 + 5x - 2 over the interval [2, 5]. This problem covers important calculus concepts, including integration and applying the fundamental theorem of calculus. Suitable for Grades 11-12.

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