Let's proceed step-by-step to evaluate the given integrals using the provided information.
### Part a: [tex]\(\int_1^3 6 f(x) \, dx\)[/tex]
We are given [tex]\(\int_1^3 f(x) \, dx = 5\)[/tex].
To evaluate [tex]\(\int_1^3 6 f(x) \, dx\)[/tex], we can use the property of integrals that states if [tex]\(c\)[/tex] is a constant, then:
[tex]\[
\int_a^b c \cdot f(x) \, dx = c \cdot \int_a^b f(x) \, dx
\][/tex]
Here, [tex]\(c = 6\)[/tex], [tex]\(a = 1\)[/tex], and [tex]\(b = 3\)[/tex]. Substituting these values, we get:
[tex]\[
\int_1^3 6 f(x) \, dx = 6 \cdot \int_1^3 f(x) \, dx
\][/tex]
Using the given value [tex]\(\int_1^3 f(x) \, dx = 5\)[/tex], we substitute it in:
[tex]\[
\int_1^3 6 f(x) \, dx = 6 \cdot 5 = 30
\][/tex]
Thus, the answer for part (a) is:
[tex]\[
\int_1^3 6 f(x) \, dx = 30
\][/tex]
### Part b: [tex]\(\int_3^8 -9 g(x) \, dx\)[/tex]
We are given [tex]\(\int_3^8 g(x) \, dx = 6\)[/tex].
To evaluate [tex]\(\int_3^8 -9 g(x) \, dx\)[/tex], we use a similar property of integrals as in part (a):
[tex]\[
\int_a^b c \cdot g(x) \, dx = c \cdot \int_a^b g(x) \, dx
\][/tex]
Here, [tex]\(c = -9\)[/tex], [tex]\(a = 3\)[/tex], and [tex]\(b = 8\)[/tex]. Substituting these values, we get:
[tex]\[
\int_3^8 -9 g(x) \, dx = -9 \cdot \int_3^8 g(x) \, dx
\][/tex]
Using the given value [tex]\(\int_3^8 g(x) \, dx = 6\)[/tex], we substitute it in:
[tex]\[
\int_3^8 -9 g(x) \, dx = -9 \cdot 6 = -54
\][/tex]
Thus, the answer for part (b) is:
[tex]\[
\int_3^8 -9 g(x) \, dx = -54
\][/tex]
In summary, the evaluated integrals are:
- For part (a): [tex]\(\int_1^3 6 f(x) \, dx = 30\)[/tex]
- For part (b): [tex]\(\int_3^8 -9 g(x) \, dx = -54\)[/tex]
