calculus - Why does an anti-derivative "preserve" a translation from the function being integrated?

To illustrate my question, consider the function given by the following equation:

$f(x)=36(x-8)^{11}$

My AP Calculus AB teacher says the anti-derivative is $F(x) = 3(x-8)^{12}+C$. I understand easily using the reverse power rule that if we simply had $g(x)=36x^{11}$, then we could easily use the reverse power rule as well as the constant multiple rule to get an anti-derivative of $G(x) = 3x^{12}+C$. But what to do about the translation -- and why does what the teacher did work?

I know from class that $[f(x+a)]' = f'(x+a)$, which was justified to us graphically and seemed to make sense. But that is a differentiation rule, not an integration rule. Obviously, the two operations are related, but I've been itching to see a more formal proof of why the "horizontal translation" rule apparently holds in both cases... IF it universally holds, of course -- maybe it's in fact specific to certain types of functions, and just happens to hold in the particular problem my teacher gave us!

Is anyone willing to attempt to justify why what my teacher did to find the anti-derivative works, in a satisfying way? (The chain rule is, I suppose, okay if you must use it, although a proof without would be better because I haven't really learned the chain rule much yet.)

Answer

It really is as simple as, if $ g'(x) = h(x) $, then $ \int{h(x)dx} = g(x) $.

Now, in the above sentence, replace $ g(x) $ with $ f(x-c) $ and $ h(x) $ with $ f'(x-c) $. Theres nothing more to say.