Shin C. answered • 05/11/20

UCLA Undergrad Expert/Enthusiast in Calculus, K-12, and SAT Math

Hi Katherine! Good luck on your AP Calculus AB exam (I'm taking BC on Tuesday too :) )

So to find ∫ f " (2x) dx, let's make this as simple as possible!

You should know, from the implications of the Fundamental Theorem of Calculus that:

∫ f ' (x) dx = f (x)

By the same logic then, ∫ f " (x) dx = f ' (x)

Also, let u = 2x. By u-substitution, du = 2 dx, so dx = du / 2

Plugging those into the integral above yields: (1/2) * ∫ f " (u) du = 1/2 * f ' (u) = 0.5 * f ' (2x).

I understand why you might think of the antiderivative of 2x is x ^ 2 because of how that's the antiderivative. BUT the function to integrate IS NOT 2x, because the function that we ought to integrate is f " (2x).

I hope this helped! Please let me know if you would like additional assistance! Thanks!

Katherine W.

Oh okay yes, that makes much more sense! Thank you so much!! Good luck to you as well! :)05/11/20