To solve for [tex]\(\int_2^5 3g(x) \, dx\)[/tex], let's proceed step-by-step using the given integrals:
1. We know [tex]\(\int_2^6 g(x) \, dx = 12\)[/tex].
2. We also know [tex]\(\int_5^6 g(x) \, dx = -3\)[/tex].
By the properties of definite integrals, we can break the integral of [tex]\(g(x)\)[/tex] over the interval from 2 to 6 into two pieces:
[tex]\[
\int_2^6 g(x) \, dx = \int_2^5 g(x) \, dx + \int_5^6 g(x) \, dx
\][/tex]
Substituting the given values, we get:
[tex]\[
12 = \int_2^5 g(x) \, dx + (-3)
\][/tex]
Solving for [tex]\(\int_2^5 g(x) \, dx\)[/tex], we add 3 to both sides:
[tex]\[
12 + 3 = \int_2^5 g(x) \, dx
\][/tex]
[tex]\[
\int_2^5 g(x) \, dx = 15
\][/tex]
Next, we need to find [tex]\(\int_2^5 3g(x) \, dx\)[/tex]. By the linearity property of integrals, we can factor out the constant 3:
[tex]\[
\int_2^5 3g(x) \, dx = 3 \int_2^5 g(x) \, dx
\][/tex]
Substituting the value of [tex]\(\int_2^5 g(x) \, dx\)[/tex]:
[tex]\[
\int_2^5 3g(x) \, dx = 3 \times 15
\][/tex]
[tex]\[
\int_2^5 3g(x) \, dx = 45
\][/tex]
Thus, the value of [tex]\(\int_2^5 3g(x) \, dx\)[/tex] is 45.
