(\int f(x), x, 0, 9) = 4 then (\int f(3x), x, 0 ,3) =
Equals 4/3 but how do you do it using antidifferentiation and not by showing examples?
You are given that $\displaystyle \int_0^9 f(x)dx= 4$ which, since we can always change a dummy variable, is the same as $\displaystyle \int_0^9 f(u)du= 4$. To integrate $\displaystyle \int_0^3 f(3x)dx$, let u= 3x so x= u/3 as TheEmptySet suggested. Then dx= (1/3)du. When x= 0, u= 0. When x= 3, u= 9. The integral becomes
$\displaystyle \int_0^9 f(u)((1/3)du)= \frac{1}{3}\int_0^9 f(u)du$