Question

If $\int f\left(x\right)dx=F\left(x\right)$, then $\int f\left(5x\right)dx=\frac{F\left(5x\right)}{5}+C$

• A. True
• B. False

Answer 1

The correct option is A

True

$\int f\left(5x\right)dx$ 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).

We can verify the given statement by taking derivative of $\frac{F\left(5x\right)}{5}$ and checking if we get f(5x).

Derivative of $\frac{F\left(5x\right)}{5}=\frac{1}{5}\times \frac{d}{dx}\left(F\right(5x\left)\right)=\frac{1}{5}\times 5F^{\prime }\left(5x\right)=f\left(5x\right)$ . So, the given statement is true.

We can verify the same by using the theorem on integration also.

It says If $\int f\left(x\right)=F\left(x\right)$, then $\int f(ax+b)=\frac{F(ax+b)}{a}+c$

In the above question a = 5 and b = 0. By substituting this in the expression we will get $\int f\left(5x\right)dx=\frac{F\left(5x\right)}{5}$