To find the value of [tex]\(\int_2^5 3g(x) \, dx\)[/tex], let's follow these steps in a detailed manner.
1. Begin with the given integral [tex]\(\int_2^5 g(x) \, dx = 12\)[/tex].
2. Recognize that the integral [tex]\(\int_2^5 3g(x) \, dx\)[/tex] involves a constant multiplier of 3 inside the integrand. Integral properties allow us to factor out constants. This property can be expressed as:
[tex]\[
\int_a^b c \cdot f(x) \, dx = c \cdot \int_a^b f(x) \, dx
\][/tex]
where [tex]\(c\)[/tex] is a constant.
3. Applying this property to our specific case:
[tex]\[
\int_2^5 3g(x) \, dx = 3 \cdot \int_2^5 g(x) \, dx
\][/tex]
4. Substitute the given integral value into the equation:
[tex]\[
\int_2^5 3g(x) \, dx = 3 \cdot 12
\][/tex]
5. Calculate the product:
[tex]\[
3 \cdot 12 = 36
\][/tex]
Thus, the value of [tex]\(\int_2^5 3g(x) \, dx\)[/tex] is 36. Therefore, the correct answer is not among the options provided.
Given the computations and results, it seems there might be an issue with the options listed. The correct calculation yields an answer of 36, which does not match any of the given choices (45, 15, 5, 3).
Please verify if there is possibly a typo or error in the provided options.
