Calculate the internal \, assuming that internal 5,0 f(x)dx=5. What is internal 5,0 (f(x)-5x)dx

To solve the problem, we can use the linearity property of definite integrals, which allows us to separate the integral into two parts. The problem is given as:

\[
\text{Given } \int_{0}^{5} f(x) \, dx = 5
\]

We need to calculate:

\[
\int_{0}^{5} \left(f(x) - 5x\right) \, dx
\]

This can be split into two integrals:

\[
\int_{0}^{5} \left(f(x) - 5x\right) \, dx = \int_{0}^{5} f(x) \, dx - \int_{0}^{5} 5x \, dx
\]

We already know that:

\[
\int_{0}^{5} f(x) \, dx = 5
\]

Now, we calculate the second integral:

\[
\int_{0}^{5} 5x \, dx
\]

The integral of \(5x\) with respect to \(x\) is:

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

Now, evaluate this definite integral from 0 to 5:

\[
\int_{0}^{5} 5x \, dx = \left[\frac{5x^2}{2}\right]_{0}^{5} = \frac{5(5)^2}{2} - \frac{5(0)^2}{2} = \frac{5 \times 25}{2} = \frac{125}{2} = 62.5
\]

Finally, subtract this result from the first integral:

\[
\int_{0}^{5} \left(f(x) - 5x\right) \, dx = 5 - 62.5 = -57.5
\]

So, the value of the integral is:

\[
\int_{0}^{5} \left(f(x) - 5x\right) \, dx = -57.5
\]

Would you like more details, or do you have any questions? 

Here are some related questions:
1. What are the basic properties of definite integrals?
2. How do you calculate definite integrals for polynomial functions?
3. What is the geometric interpretation of definite integrals?
4. How does the linearity property of integrals help in solving complex integrals?
5. What are some common mistakes to avoid when calculating definite integrals?

**Tip:** Always double-check your limits of integration and ensure that you evaluate the definite integral correctly at both bounds.

Learn how to solve a definite integral problem involving the linearity property of integrals. This example demonstrates how to calculate ∫[0,5] (f(x) - 5x) dx given that ∫[0,5] f(x) dx = 5. Suitable for students in Grades 11-12.

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