https://r2.erweima.ai/mathbot/image/172491462389178x7t3it.jpg

The equation shown in the image is:

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

You want to find the definite integral of \( a = 5x \) from 1 to 2.

Let's calculate it:

Given \( a = 5x \), the integral is:

\[
\int_{1}^{2} 5x \, dx
\]

To solve this, integrate \( 5x \) with respect to \( x \):

\[
\int 5x \, dx = \frac{5x^2}{2} + C
\]

Now, evaluate this definite integral from 1 to 2:

\[
\left[\frac{5x^2}{2}\right]_{1}^{2} = \frac{5(2^2)}{2} - \frac{5(1^2)}{2} = \frac{5 \times 4}{2} - \frac{5 \times 1}{2} = \frac{20}{2} - \frac{5}{2} = 10 - 2.5 = 7.5
\]

So, the value of the definite integral from 1 to 2 is **7.5**.

Would you like more details or have any questions?

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Here are five related questions:

1. What is the indefinite integral of \( 2x^3 \)?
2. How do you solve definite integrals with polynomial functions?
3. What is the physical interpretation of a definite integral?
4. How does changing the limits of integration affect the integral's value?
5. How would you compute the area under the curve of a function \( f(x) \)?

**Tip:** Always ensure to correctly substitute the limits of integration after finding the indefinite integral when evaluating definite integrals.

Learn how to calculate the definite integral of the function 5x from 1 to 2 with a detailed step-by-step solution. This problem covers key concepts in calculus, including definite integration and evaluating integrals. Suitable for advanced high school students.

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