They all come from various substitution techniques and integration by parts. The techniques should have been taught to you, at least I would hope. It is in my opinion unreasonable to flash-card your way through the class, though the first time I took the class, I was guilty of attempting it.
Example integral off of the top of my head:
\(\displaystyle \displaystyle \int \dfrac{1}{1+x^2}dx\)
This should be recognizable as a trigonometric-substitution problem. Since you know \(\displaystyle 1+\tan^2t=\sec^2t\), making the substitution \(\displaystyle x=\tan t \) simplifies your integral to
\(\displaystyle \displaystyle \int dt = t+C = \tan^{-1}(x)+C\)
Now it might seem easier to remember, "oh, thats arctan of x", but such substitutions encompass a large amount of the integration formulas you see. Learn how they came to be, and you will understand them (and possibly assimilate many of them in the process) rather than trying to stuff your brain.
