Chapter 5, Problem 15RE

(a)

To determine

To evaluate: The value of integral ${\displaystyle \underset{-4}{\overset{4}{\int }}f\left(x\right)dx}$ by using the facts ${\displaystyle \underset{0}{\overset{4}{\int }}f\left(x\right)dx}=10$ and ${\displaystyle \underset{0}{\overset{4}{\int }}g\left(x\right)dx}=20$ where f is an even function and g is an odd function.

(b)

To determine

To evaluate: The value of integral ${\displaystyle \underset{-4}{\overset{4}{\int }}3g\left(x\right)dx}$ by using the facts ${\displaystyle \underset{0}{\overset{4}{\int }}f\left(x\right)dx}=10$ and ${\displaystyle \underset{0}{\overset{4}{\int }}g\left(x\right)dx}=20$ where f is an even function and g is an odd function.

(c)

To determine

To evaluate: The value of integral ${\displaystyle \underset{-4}{\overset{4}{\int }}\left(4f\left(x\right)-3g\left(x\right)\right)dx}$ by using the facts ${\displaystyle \underset{0}{\overset{4}{\int }}f\left(x\right)dx}=10$ and ${\displaystyle \underset{0}{\overset{4}{\int }}g\left(x\right)dx}=20$ where f is an even function and g is an odd function.

(d)

To determine

To evaluate: The value of integral ${\displaystyle \underset{0}{\overset{1}{\int }}8xf\left(4x^2\right)dx}$ by using the facts ${\displaystyle \underset{0}{\overset{4}{\int }}f\left(x\right)dx}=10$ and ${\displaystyle \underset{0}{\overset{4}{\int }}g\left(x\right)dx}=20$ where f is an even function and g is an odd function.

e.

To determine

To evaluate: The value of integral ${\displaystyle \underset{-2}{\overset{2}{\int }}3xf\left(x\right)dx}$ by using the facts ${\displaystyle \underset{0}{\overset{4}{\int }}f\left(x\right)dx}=10$ and ${\displaystyle \underset{0}{\overset{4}{\int }}g\left(x\right)dx}=20$ where f is an even function and g is an odd function.