1. ## Integration

(\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?

2. let

$u=\frac{x}{3} \implies du=\frac{dx}{3}$

and don't forget the variable of integration is a dummy variable.

try to sub this into the first integral.

3. so does that mean that 1/3 applies to the first integral. And also the integral is 0, 9 on the first integral and 0,3 on the second..... how does that work out?

Originally Posted by TheEmptySet
let

$u=\frac{x}{3} \implies du=\frac{dx}{3}$

and don't forget the variable of integration is a dummy variable.

try to sub this into the first integral.

4. and bring the 3 out from the 3x?

5. You are given that $\int_0^9 f(x)dx= 4$ which, since we can always change a dummy variable, is the same as $\int_0^9 f(u)du= 4$. To integrate $\int_0^3 f(3x)dx$, 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
$\int_0^9 f(u)((1/3)du)= \frac{1}{3}\int_0^9 f(u)du$