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.

• ${\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{kf}\left(\mathrm{x}\right)d\mathrm{x}=\_\_\_\_\_\_\_.$

Short answer

Ans:${\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{kf}\left(\mathrm{x}\right)d\mathrm{x}=\mathrm{k}{\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{kF}\left(b\right)-\mathrm{kF}\left(a\right)$

Step-by-step solution

Step 1. Given information:

${\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{kf}\left(\mathrm{x}\right)d\mathrm{x}$

Step 2. Solution:

$$\begin{gathered} {\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{kf}\left(\mathrm{x}\right)d\mathrm{x}=\mathrm{k}{\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ \mathrm{If}\mathrm{f}\mathrm{is}\mathrm{continuous}\mathrm{on}[\mathrm{a},\mathrm{b}]\mathrm{and}\mathrm{F}\mathrm{is}\mathrm{any}\mathrm{antiderivative}\mathrm{of}\mathrm{f},\mathrm{then} \\ ={\mathrm{k}\left[\mathrm{F}\left(\mathrm{x}\right)\right]}_{\mathrm{a}}^{\mathrm{b}} \\ =\mathrm{k}\left(\mathrm{F}\left(b\right)-\mathrm{F}\left(a\right)\right) \\ =\mathrm{kF}\left(b\right)-\mathrm{kF}\left(a\right) \end{gathered}$$