Update: For what it's worth, I will wait another few hours to see if anyone else has a more comprehensive answer to my question. But if not, I will "accept" one of the two extant answers, both of which are very good although not quite as comprehensive as I had hoped.
I am taking AP Calculus AB right now and we are learning about anti-derivatives (indefinite integrals) for Unit II. Before that, we learned some basic derivative rules for transformed functions, such as:
$ [f (x + a)]]' = f'(x + a)$
(We also learned the derivatives of some elementary functions, e.g. polynomials, exponential functions, sine, and cosine.)
For anti-derivatives we have likewise memorized (and proved) formulas for some basic functions, but unlike with derivatives we have not been taught very much at all about what to do with transformed functions.
Consider, say, finding the anti-derivative $F(x)$ if $f(x)=1/(4x)$. Or finding the anti-derivative $G(x)$ if $g(x)=\cos(4x)$. (Or even worse, how about if $f(x)$ was actually $1/(4x-3)$, and $g(x)$ was actually $\cos(4x-3)$?) I am not entirely sure how to systematically and carefully go about solving such problems.
Should I try to learn integral u-substitution or any tricks like that (which we haven't covered in class yet), or am I better off just trying to intuitively "reverse" the differentiation rules as best I can? I want to figure out a relatively efficient method of integrating basic functions but right now am pretty confused. (Often attempting to reverse the differentiation rules kinda gives me a headache haha and I get utterly lost because it's hard to think about things backwards.)