Question

How would I do this?

Answer 1

sub \(u=2x\)

Answer 2

Hmm... but I don't have the function... Im kind of confused

Answer 3

\[\int\limits_0^2 f(2x)\,dx~~\stackrel{u=2x}{=}~~ \frac{1}{2}\int\limits_0^4 f(u)\,du = \frac{1}{2}*20=10\]

Answer 4

Why does the limit of integration change from 4 to 2?

Answer 5

scratch that, lets work it again from beginning

Answer 6

you want to evaluate
\[\int\limits_0^2f(2x)\, dx\]

substitute \(u=2x \implies du=2dx\implies dx=\dfrac{du}{2}\)

Answer 7

next work the bounds,
as \(x\to 0\), what does \(u\to\) ?
as \(x\to 2\), what does \(u\to\) ?

Answer 8

1) x -> 0, u -> 0
2) x -> 2, u -> 4

Answer 9

so upon substitution, bounds change from (0, 2) to (0, 4)
and the differential changes from dx to du/2

plug them in

Answer 10

okay, so then that's \[\int\limits_{0}^{4}f(u) \frac{ du }{ 2 }\]
and then the 1/2 comes out of the integral

Answer 11

Yes, next recall that the variable in definite integral is "dummy"
\[\int\limits_a^b f(\color{red}{x})\,d\color{red}{x} = \int\limits_a^b f(\color{red}{t})\,d\color{red}{t}=\int\limits_a^b f(\color{red}{\spadesuit})\,d\color{red}{\spadesuit}\]

Answer 12

right

Answer 13

Ohhh now I get what's happening!

Answer 14

@ganeshie8 Thanks so much!