3.4b Laws of Definite Integrals

3.4b Laws of Definite Integrals



Example:
Given that ${\displaystyle {\int }_3^{7}f(x)dx=5}$ , find the values for each of the following:

$\begin{array}{l}\text{(a)}{\displaystyle {\int }_3^{7}6f(x)dx}\\ \text{(b)}{\displaystyle {\int }_3^7[3-f(x)]dx}\\ \text{(c)}{\displaystyle {\int }_7^{3}2f(x)dx}\\ \text{(d)}{\displaystyle {\int }_3^{4}f(x)dx}+{\displaystyle {\int }_4^{5}f(x)dx+{\displaystyle {\int }_3^{7}f(x)dx}}\\ \text{(e)}{\displaystyle {\int }_3^7\frac{f(x)+7}{2}dx}\end{array}$


Solution:
$\begin{array}{l}\text{(a)}{\displaystyle {\int }_3^{7}6f(x)dx}=6{\displaystyle {\int }_3^{7}f(x)dx}\\ =6(5)=30\\ \\ \text{(b)}{\displaystyle {\int }_3^7[3-f(x)]dx={\displaystyle {\int }_3^{7}3dx-{\displaystyle {\int }_3^{7}f(x)}dx}}\\ ={\left[3x\right]}_3^7-5\\ =\left[3(7)-3(3)\right]-5\\ =7\\ \\ \text{(c)}{\displaystyle {\int }_7^{3}2f(x)dx}=-{\displaystyle {\int }_3^{7}2f(x)dx}\\ =-2{\displaystyle {\int }_3^{7}f(x)dx}\\ =-2(5)\\ =-10\\ \\ \text{(d)}{\displaystyle {\int }_3^{4}f(x)dx}+{\displaystyle {\int }_4^{5}f(x)dx+{\displaystyle {\int }_3^{7}f(x)dx}}\\ ={\displaystyle {\int }_3^{7}f(x)dx}\\ =5\\ \\ \text{(e)}{\displaystyle {\int }_3^7\frac{f(x)+7}{2}dx}={\displaystyle {\int }_3^7\left[\frac{1}{2}f(x)+\frac{7}{2}\right]dx}\\ ={\displaystyle {\int }_3^7\frac{1}{2}f(x)dx}+{\displaystyle {\int }_3^7\frac{7}{2}dx}\\ =\frac{1}{2}{\displaystyle {\int }_3^{7}f(x)dx}+{\left[\frac{7x}{2}\right]}_3^7\\ =\frac{1}{2}(5)+\left[\frac{7(7)}{2}-\frac{7(3)}{2}\right]\\ =\frac{5}{2}+14\\ =16\frac{1}{2}\end{array}$