Suppose Integral from 2 to 4 f left parenthesis x right parenthesis dx∫24f(x)dxequals=44​, Integral from 4 to 7 f left parenthesis x right parenthesis dx∫47f(x)dxequals=negative 5−5​, and Integral from 4 to 7 g left parenthesis x right parenthesis dx∫47g(x)dxequals=22. Evaluate the integrals in parts a dash– d.
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Part 1
a. Integral from 2 to 4 5 f left parenthesis x right parenthesis dx∫245f(x)dxequals=2020 ​(Simplify your​ answer.)
Part 2
b. Integral from 4 to 7 minus 6 g left parenthesis x right parenthesis dx∫47−6g(x)dxequals=enter your response here ​(Simplify your​ answer.)

Let's solve the integrals for parts (a) and (b) using the information provided.

### Part (a)
We are given:
\[
\int_{2}^{4} f(x) \, dx = 44
\]
We need to evaluate:
\[
\int_{2}^{4} 5f(x) \, dx
\]
Since the integral of a constant times a function can be expressed as the constant times the integral of the function:
\[
\int_{2}^{4} 5f(x) \, dx = 5 \times \int_{2}^{4} f(x) \, dx = 5 \times 44 = 220
\]

### Part (b)
We are given:
\[
\int_{4}^{7} g(x) \, dx = 22
\]
We need to evaluate:
\[
\int_{4}^{7} -6g(x) \, dx
\]
Similarly, using the property of integrals, we can factor out the constant:
\[
\int_{4}^{7} -6g(x) \, dx = -6 \times \int_{4}^{7} g(x) \, dx = -6 \times 22 = -132
\]

### Final Answers
- **(a) \(\int_{2}^{4} 5f(x) \, dx = 220\)**
- **(b) \(\int_{4}^{7} -6g(x) \, dx = -132\)**

Would you like more details or have any questions? Here are 5 related questions to deepen your understanding:

1. How would you evaluate \(\int_{2}^{7} (f(x) + g(x)) \, dx\) using the provided integrals?
2. What is the integral of a sum of functions over the same interval?
3. Can you explain why we can factor out constants in definite integrals?
4. What happens if the integrand is multiplied by a variable instead of a constant?
5. How do the properties of definite integrals relate to the linearity of integration?

**Tip:** Remember, the linearity of integration allows you to split sums and factor out constants, making it easier to handle complex integrals by breaking them down into simpler parts.

Suppose Integral from 2 to 4 f(x) dx = 44, Integral from 4 to 7 f(x) dx = -5, and Integral from 4 to 7 g(x) dx = 22. Evaluate the integrals in parts a–d.
Part 1
a. Integral from 2 to 4 5f(x) dx (Simplify your answer.)
Part 2
b. Integral from 4 to 7 -6g(x) dx (Simplify your answer.)

Learn how to evaluate definite integrals involving constant multiplication, using properties of integration. Solve integrals such as ∫2 to 4 5f(x) dx and ∫4 to 7 -6g(x) dx step-by-step. Suitable for students in Grades 11-12.

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