Chapter 5, Problem 10RWDT

To determine

The relationship between the given functions.

Answer to Problem 10RWDT

It has been determined that the required relationship between the given functions is:

  $g\left(x\right)={\displaystyle {\int }_0^{2}f\left(t\right)dt}+j\left(x\right)$

Explanation of Solution

Given:

The functions, $g\left(x\right)={\displaystyle {\int }_0^{x}f\left(t\right)dt}$ and $j\left(x\right)={\displaystyle {\int }_2^{x}f\left(t\right)dt}$ .

Concept used:

According to the property of definite integration:

  ${\displaystyle {\int }_a^{b}f\left(t\right)dt}={\displaystyle {\int }_a^{c}f\left(t\right)dt}+{\displaystyle {\int }_c^{b}f\left(t\right)dt}$ for any $a\le c\le b$ .

Calculation:

As discussed, ${\displaystyle {\int }_a^{b}f\left(t\right)dt}={\displaystyle {\int }_a^{c}f\left(t\right)dt}+{\displaystyle {\int }_c^{b}f\left(t\right)dt}$ for any $a\le c\le b$ .

Put $a=0$ , $b=x$ and $c=2$ in the above expression to get,

  ${\displaystyle {\int }_0^{x}f\left(t\right)dt}={\displaystyle {\int }_0^{2}f\left(t\right)dt}+{\displaystyle {\int }_2^{x}f\left(t\right)dt}$

Put $g\left(x\right)={\displaystyle {\int }_0^{x}f\left(t\right)dt}$ and $j\left(x\right)={\displaystyle {\int }_2^{x}f\left(t\right)dt}$ in the above equation to get,

  $g\left(x\right)={\displaystyle {\int }_0^{2}f\left(t\right)dt}+j\left(x\right)$

This is the required relationship between the given functions.

Conclusion:

It has been determined that the required relationship between the given functions is:

  $g\left(x\right)={\displaystyle {\int }_0^{2}f\left(t\right)dt}+j\left(x\right)$