Q. 75 - Calculus

Question

Use the Fundamental Theorem of Calculus to give alternative proofs of the integration facts shown in Exercises 72–76. You may assume that all functions here are integrable 

${\int }_a^{b}kf\left(x\right) dx=k{\int }_a^{b}f\left(x\right)$

Step-by-Step Solution

Answer

Proved that ${\int }_a^{b}kf\left(x\right) dx=k{\int }_a^{b}f\left(x\right)$

Step 1. Given information

The given integral ${\int }_a^{b}kf\left(x\right) dx=k{\int }_a^{b}f\left(x\right)$

Step 2. Prove that ∫ a b k f ( x )   d x = k ∫ a b f ( x ) using the fundamental theorem of calculas

$$\begin{gathered} {\int }_a^{b}kf\left(x\right) dx=k{\left[f\left(x\right)\right]}_a^b [u\sin g fundamental theorem] \\ =k\left(f\left(b\right)\right)-k\left(f\left(a\right)\right) \\ =k\left(f\left(a\right)-f\left(b\right)\right) \\ =k{\left[F\left(x\right)\right]}_a^b \\ =k{\int }_a^{b}f\left(x\right) dx \end{gathered}$$

Therefore,it is proved that ${\int }_a^{b}kf\left(x\right) dx=k{\int }_a^{b}f\left(x\right)$