MHB True or False Integral Calculus Question #2

[MermaidWonders]

True or False: Let $F(x)$ be an antiderivative of a function $f(x)$. Then, $F(2x)$ is an antiderivative of the function $f(2x)$.

 

[Greg]

What is the derivative of $F(2x)$? Or, perhaps more to the point, what do you get if you integrate $f(2x)$ using the substitution $u=2x,du=2\,dx$?

 

[MermaidWonders]

Kinda confused... how would you do that using substitution?

 

[Greg]

Try it and post your effort. :)

 

[MermaidWonders]

OK... Here's what I came up with... took me a while but I'm still kinda confused with myself. :(

Let $u = 2x$. Then $du/dx = 2$ --> $dx = du/2$.

$\int f(2x)dx$ then becomes $\int f(u)(du/2) = (1/2)f(u)du = (1/2)F(u) = (1/2)F(2x)$.

 

[MermaidWonders]

So I'm guessing that's false because $\int f(2x)dx \ne F(2x)$ but $(\frac{1}{2})F(2x)$, right?

 

[Greg]

OK... Here's what I came up with... took me a while but I'm still kinda confused with myself. :(

Let $u = 2x$. Then $du/dx = 2$ --> $dx = du/2$.

$\int f(2x)dx$ then becomes $\int f(u)(du/2) = (1/2)f(u)du = (1/2)F(u) = (1/2)F(2x)$.

That's right. Chain rule. :)

Excellent work!

 

[MermaidWonders]

Alright, thanks so much! :)