Chapter 5, Problem 1RQ

To determine

Whether the given statement is true or false.

Answer to Problem 1RQ

The statement ${\displaystyle \underset{a}{\overset{b}{\int }}\left[f\left(x\right)+g\left(x\right)\right]}dx={\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx+}{\displaystyle \underset{a}{\overset{b}{\int }}g\left(x\right)dx}$ is true.

Explanation of Solution

Given information:

If f and g are continuous on $\left[a,b\right]$, then the function as follows:

${\displaystyle \underset{a}{\overset{b}{\int }}\left[f\left(x\right)+g\left(x\right)\right]}dx={\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx+}{\displaystyle \underset{a}{\overset{b}{\int }}g\left(x\right)dx}$ (1)

Explanation for justification of statement:

Show the basic property of integral as follows:

${\displaystyle \underset{a}{\overset{b}{\int }}\left[f\left(x\right)+g\left(x\right)\right]}dx={\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx+}{\displaystyle \underset{a}{\overset{b}{\int }}g\left(x\right)dx}$ (2)

The integral property is valid if f and g are continuous functions.

The given statement is same as the property of integral. Hence, the statement is true.

Therefore, the statement ${\displaystyle \underset{a}{\overset{b}{\int }}\left[f\left(x\right)+g\left(x\right)\right]}dx={\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx+}{\displaystyle \underset{a}{\overset{b}{\int }}g\left(x\right)dx}$ is true.