Answer
a. $4$
b. $2$
c. $-2$
d. $-2\pi$,
e. $\frac{8}{5}$
Work Step by Step
Given $\int_{-2}^23f(x)dx=3\int_{-2}^2f(x)dx=12$, $\int_{-2}^5f(x)dx=6$, and $\int_{-2}^5g(x)dx=2$, we have
a. $\int_{-2}^2f(x)dx=12/3=4$
b. $\int_{2}^5f(x)dx=\int_{-2}^5f(x)dx-\int_{-2}^2f(x)dx=6-4=2$
c. $\int_{5}^{-2}g(x)dx=-\int_{-2}^5g(x)dx=-2$
d. $\int_{-2}^5(-\pi g(x))dx=-\pi\int_{-2}^5 g(x)dx=-2\pi$,
e. $\int_{-2}^5(\frac{f(x)+g(x)}{5})dx=\int_{-2}^5(\frac{f(x)}{5})dx+\int_{-2}^5(\frac{g(x)}{5})dx=\frac{1}{5}\int_{-2}^5f(x)dx+\frac{1}{5}\int_{-2}^5g(x)dx=\frac{6}{5}+\frac{2}{5}=\frac{8}{5}$
