Antiderivative Help!

Question

So I am preparing for AP calculus AB exam coming up, and I recently am stuck on this minuscule problem that frequently shows up in other forms.

If I am told to Integrate f’’(2x) , what should the antiderivative be? Their answers show it is just [1/2 f’(2x)] which in my mind doesn’t make sense because, where is the ^2 above? And wouldn’t it just be (2x^2)/2, which is therefore just x^2?

Any thoughts on this would be very much appreciated! :)

Answer 1

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!