Q. 27 - Calculus

Question

The algebra of definite integrals:

Fill in the blanks to complete the definite integral rules that follow. You may assume that f and g are integrable functions on $[a,b]$, that $c\in [a,b]$, and that k is any real number.

• ${\int }_a^b\left(f\right(x)+g(x\left)\right) dx =\_\_\_\_\_\_\_\_\_.$
  

Step-by-Step Solution

Answer

Ans: ${\int }_a^b\left(f\right(x)+g(x\left)\right) dx ={\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}+{\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{F}\left(b\right)-\mathrm{F}\left(a\right)+\mathrm{G}\left(b\right)-\mathrm{G}\left(a\right)$

Step 1. Given information:

${\int }_a^b\left(f\right(x)+g(x\left)\right) dx$

Step 2 .Solution:

$$\begin{gathered} {\int }_{\mathrm{a}}^{\mathrm{b}}\left(\mathrm{f}\right(\mathrm{x})+\mathrm{g}(\mathrm{x}\left)\right)d\mathrm{x}={\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}+{\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx} \\ \mathrm{If} \mathrm{f} \mathrm{is} \mathrm{continuous} \mathrm{on} [\mathrm{a}, \mathrm{b}] \mathrm{and} \mathrm{F},\mathrm{G} \mathrm{is} \mathrm{any} \mathrm{anti}-\mathrm{derivative} \mathrm{of} \mathrm{f} , \mathrm{then} \\ ={\left[\mathrm{F}\left(\mathrm{x}\right)\right]}_{\mathrm{a}}^{\mathrm{b}}+{\left[\mathrm{G}\left(\mathrm{x}\right)\right]}_{\mathrm{a}}^{\mathrm{b}} \\ =\left(\mathrm{F}\left(b\right)-\mathrm{F}\left(a\right)\right)+\left(\mathrm{G}\left(b\right)-\mathrm{G}\left(a\right)\right) \\ =\mathrm{F}\left(b\right)-\mathrm{F}\left(a\right)+\mathrm{G}\left(b\right)-\mathrm{G}\left(a\right) \\ =\mathrm{F}(b-a)+\mathrm{G}(b-a) \\ =(b-a)[\mathrm{F}+\mathrm{G}] \end{gathered}$$