Let's proceed with evaluating the given integrals step-by-step based on the provided information:
### Given:
1. [tex]\(\int_1^3 f(x) \, dx = 5\)[/tex]
2. [tex]\(\int_3^8 f(x) \, dx = -4\)[/tex]
3. [tex]\(\int_3^8 g(x) \, dx = 6\)[/tex]
### Part a: [tex]\(\int_1^3 6 f(x) \, dx\)[/tex]
To find [tex]\(\int_1^3 6 f(x) \, dx\)[/tex], we use the property of integrals which states that multiplying the integrand by a constant can be factored out:
[tex]\[
\int_1^3 6 f(x) \, dx = 6 \cdot \int_1^3 f(x) \, dx
\][/tex]
Substitute the given value for [tex]\(\int_1^3 f(x) \, dx\)[/tex]:
[tex]\[
\int_1^3 6 f(x) \, dx = 6 \cdot 5 = 30
\][/tex]
### Part b: [tex]\(\int_3^8 -9 g(x) \, dx\)[/tex]
For [tex]\(\int_3^8 -9 g(x) \, dx\)[/tex], again, we use the property of integrals involving a constant multiple:
[tex]\[
\int_3^8 -9 g(x) \, dx = -9 \cdot \int_3^8 g(x) \, dx
\][/tex]
Substitute the given value for [tex]\(\int_3^8 g(x) \, dx\)[/tex]:
[tex]\[
\int_3^8 -9 g(x) \, dx = -9 \cdot 6 = -54
\][/tex]
### Part c: [tex]\(\int_3^8 [9 f(x) - g(x)] \, dx\)[/tex]
To evaluate [tex]\(\int_3^8 [9 f(x) - g(x)] \, dx\)[/tex], we can use the linearity property of integrals:
[tex]\[
\int_3^8 [9 f(x) - g(x)] \, dx = \int_3^8 9 f(x) \, dx - \int_3^8 g(x) \, dx
\][/tex]
We evaluate each integral separately:
[tex]\[
\int_3^8 9 f(x) \, dx = 9 \cdot \int_3^8 f(x) \, dx
\][/tex]
[tex]\[
\int_3^8 g(x) \, dx = \int_3^8 g(x) \, dx
\][/tex]
Now, substitute the given values:
[tex]\[
\int_3^8 9 f(x) \, dx = 9 \cdot (-4) = -36
\][/tex]
[tex]\[
\int_3^8 g(x) \, dx = 6
\][/tex]
Combine them:
[tex]\[
\int_3^8 [9 f(x) - g(x)] \, dx = -36 - 6 = -42
\][/tex]
### Summary of Results:
- a. [tex]\(\int_1^3 6 f(x) \, dx = 30\)[/tex]
- b. [tex]\(\int_3^8 -9 g(x) \, dx = -54\)[/tex]
- c. [tex]\(\int_3^8 [9 f(x) - g(x)] \, dx = -42\)[/tex]
Final answer for part c:
[tex]\[
\int_3^8 [9 f(x) - g(x)] \, dx = -42
\][/tex]
