Let's solve each part step-by-step using the given values for the integrals [tex]\(\int_1^3 f(x) \, dx = 5\)[/tex], [tex]\(\int_3^8 f(x) \, dx = -4\)[/tex], and [tex]\(\int_3^8 g(x) \, dx = 6\)[/tex].
### Part a: [tex]\(\int_1^3 6f(x) \, dx\)[/tex]
The integral [tex]\(\int_1^3 6f(x) \, dx\)[/tex] can be simplified using the linearity property of integrals:
[tex]\[
\int_1^3 6f(x) \, dx = 6 \int_1^3 f(x) \, dx
\][/tex]
Given that [tex]\(\int_1^3 f(x) \, dx = 5\)[/tex]:
[tex]\[
6 \int_1^3 f(x) \, dx = 6 \times 5 = 30
\][/tex]
Thus, [tex]\(\int_1^3 6f(x) \, dx = 30\)[/tex].
### Part b: [tex]\(\int_3^8 -9g(x) \, dx\)[/tex]
The integral [tex]\(\int_3^8 -9g(x) \, dx\)[/tex] can also be simplified using the linearity property of integrals:
[tex]\[
\int_3^8 -9g(x) \, dx = -9 \int_3^8 g(x) \, dx
\][/tex]
Given that [tex]\(\int_3^8 g(x) \, dx = 6\)[/tex]:
[tex]\[
-9 \int_3^8 g(x) \, dx = -9 \times 6 = -54
\][/tex]
Thus, [tex]\(\int_3^8 -9g(x) \, dx = -54\)[/tex].
### Part c: [tex]\(\int_3^8 [9f(x) - g(x)] \, dx\)[/tex]
The integral [tex]\(\int_3^8 [9f(x) - g(x)] \, dx\)[/tex] can be split using the linearity property:
[tex]\[
\int_3^8 [9f(x) - g(x)] \, dx = \int_3^8 9f(x) \, dx - \int_3^8 g(x) \, dx
\][/tex]
We can simplify each part individually:
[tex]\[
\int_3^8 9f(x) \, dx = 9 \int_3^8 f(x) \, dx \quad \text{and} \quad \int_3^8 g(x) \, dx
\][/tex]
Given that [tex]\(\int_3^8 f(x) \, dx = -4\)[/tex] and [tex]\(\int_3^8 g(x) \, dx = 6\)[/tex]:
[tex]\[
9 \int_3^8 f(x) \, dx = 9 \times -4 = -36
\][/tex]
Thus,
[tex]\[
\int_3^8 [9f(x) - g(x)] \, dx = -36 - 6 = -42
\][/tex]
Thus, [tex]\(\int_3^8 [9f(x) - g(x)] \, dx = -42\)[/tex].
### Part d: [tex]\(\int_8^3 [f(x) + 7g(x)] \, dx\)[/tex]
The integral [tex]\(\int_8^3 [f(x) + 7g(x)] \, dx\)[/tex] involves reversing the limits of integration. We use the property of integrals that states if we reverse the limits, we need to multiply by [tex]\(-1\)[/tex]:
[tex]\[
\int_8^3 [f(x) + 7g(x)] \, dx = -\int_3^8 [f(x) + 7g(x)] \, dx
\][/tex]
We can now split the integral inside:
[tex]\[
\int_3^8 [f(x) + 7g(x)] \, dx = \int_3^8 f(x) \, dx + \int_3^8 7g(x) \, dx
\][/tex]
We can further simplify:
[tex]\[
\int_3^8 7g(x) \, dx = 7 \int_3^8 g(x) \, dx
\][/tex]
Given that [tex]\(\int_3^8 f(x) \, dx = -4\)[/tex] and [tex]\(\int_3^8 g(x) \, dx = 6\)[/tex]:
[tex]\[
\int_3^8 f(x) \, dx + 7 \int_3^8 g(x) \, dx = -4 + 7 \times 6 = -4 + 42 = 38
\][/tex]
Thus:
[tex]\[
\int_8^3 [f(x) + 7g(x)] \, dx = -\int_3^8 [f(x) + 7g(x)] \, dx = -38
\][/tex]
Thus, [tex]\(\int_8^3 [f(x) + 7g(x)] \, dx = -38\)[/tex].
In conclusion, the evaluated integrals are:
a. [tex]\(\int_1^3 6f(x) \, dx = 30\)[/tex]
b. [tex]\(\int_3^8 -9g(x) \, dx = -54\)[/tex]
c. [tex]\(\int_3^8 [9f(x) - g(x)] \, dx = -42\)[/tex]
d. [tex]\(\int_8^3 [f(x) + 7g(x)] \, dx = -38\)[/tex]
