Q. 76 - Calculus

Question

Prove $$\begin{gathered} \mathrm{The} \mathrm{objective} \mathrm{is} \mathrm{to} \mathrm{prove} \mathrm{that} \int kf\left(x\right) dx=k\int f\left(x\right)dx in words. \\ \mathrm{Let} \mathrm{Fx} \mathrm{be} \mathrm{the} \mathrm{derivative} \mathrm{of} f\left(x\right). \end{gathered}$$

Step-by-Step Solution

Answer

Proved

Step 1. To prove

$\int kf\left(x\right) dx=k\int f\left(x\right)dx$

Step 2. Proving

$$\begin{gathered} \int kf\left(x\right) dx=k\int f\left(x\right)dx \\ =\int kf\left(x\right) dx \\ =k{F}^{\text{'}}\left(x\right)+C \\ =k\left(F^{\text{'}}\left(x\right)+\frac{C}{k}\right) \\ =\int kf\left(x\right) dx \\ \mathrm{Hence} \mathrm{Proved} \end{gathered}$$]