Q. 78 - Calculus

Question

$$\begin{gathered} T\mathrm{he} \mathrm{functions} \mathrm{F}, \mathrm{G} \mathrm{are} \mathrm{differentiable} \mathrm{als}o F\left(x\right)=G\left(x\right). \\ \mathrm{Where}, \mathrm{C} \mathrm{is} \mathrm{any} \mathrm{constant}. \\ \mathrm{prove} \mathrm{that} G\left(x\right)= F\left(x\right) + C. \end{gathered}$$

Step-by-Step Solution

Answer

Proved

Step 1. To Prove

$G\left(x\right)= F\left(x\right) + C.$

Step 2. Proving

$$\begin{gathered} Now, \\ F\text{'}\left(x\right)=G\left(x\right) \\ \int F\left(x\right)dx=\int G\left(x\right)dx \\ F\left(x\right)+C₁=G\left(x\right)+C₂ \\ G\left(x\right)=F\left(x\right)+(C₁-C₂) \\ G\left(x\right)=F\left(x\right)+C \\ \mathrm{Therefore}, \mathrm{it} \mathrm{is} \mathrm{proved} \mathrm{tha}t G\left(x\right)=F\left(x\right)+C. \end{gathered}$$