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

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

Short answer

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

Step-by-step solution

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 ∫abkf(x) dx=k∫abf(x)using the fundamental theorem of calculas

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

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