Question

calculator allowed

1. if $f(x)=g(x)+7$ for $3 \leq x \leq 5$, then $\int_{3}^{5}f(x)+g(x)dx=$

(a) $2\int_{3}^{5}g(x)dx + 7$
(b) $2\int_{3}^{5}g(x)dx + 14$
(c) $2\int_{3}^{5}g(x)dx + 28$
(d) $\int_{3}^{5}g(x)dx + 7$
(e) $\int_{3}^{5}g(x)dx + 14$

Explanation:

Step1: Substitute \( f(x) \) into the integral

Given \( f(x) = g(x) + 7 \) for \( 3 \leq x \leq 5 \), substitute \( f(x) \) into \( \int_{3}^{5} [f(x) + g(x)] dx \).
We get \( \int_{3}^{5} [(g(x) + 7) + g(x)] dx \).

Step2: Simplify the integrand

Simplify the expression inside the integral: \( (g(x) + 7) + g(x) = 2g(x) + 7 \).
So the integral becomes \( \int_{3}^{5} (2g(x) + 7) dx \).

Step3: Use the linearity of integration

The integral of a sum is the sum of the integrals, and the integral of a constant multiple is the constant multiple of the integral. So we can split the integral:
\( \int_{3}^{5} (2g(x) + 7) dx = \int_{3}^{5} 2g(x) dx + \int_{3}^{5} 7 dx \)
\( = 2\int_{3}^{5} g(x) dx + 7\int_{3}^{5} 1 dx \)

Step4: Evaluate the integral of the constant

The integral of \( 1 \) with respect to \( x \) from \( 3 \) to \( 5 \) is \( x \big|_{3}^{5} = 5 - 3 = 2 \).
So \( 7\int_{3}^{5} 1 dx = 7\times(5 - 3) = 7\times2 = 14 \).

Step5: Combine the results

Putting it back together, we have \( 2\int_{3}^{5} g(x) dx + 14 \).

Answer:

B. \( 2\int_{3}^{5} g(x) dx + 14 \)