Question

If $\int f\left(x\right)dx=F\left(x\right)$, then $\int f\left(5x\right)dx$=

• A. $\frac{F\left(5x\right)}{5}+C$
• B. $\frac{F(x}{5}+C$
• C. $F\left(5x\right)+C$
• D. $\frac{F\left(5x\right)}{25}+C$

Answer 1

The correct option is A $\frac{F\left(5x\right)}{5}+C$

$\int f\left(5x\right)dx$ means, we want to find a function, whose derivative will give f(5x). We are given that integral of f(x) is F(x). This means, when we differentiate F(x), we will get f(x).

Derivative of $F\left(5x\right)$ is $f\left(5x\right)\times \frac{d}{dx}\left(5x\right)$ Which is $5f\left(5x\right)$

Derivative of $\frac{F\left(5x\right)}{5}=\frac{1}{5}\times \frac{d}{dx}\left(F\right(5x\left)\right)=\frac{1}{5}\times 5F^{\prime }\left(5x\right)=f\left(5x\right)$.