What I mean, is that while you can always write down a derivative for any function no matter how complicated it is using basic rules, you cannot integrate even most functions out there. Only a very small class of functions can you ever have a hope of integrating. That is why there are at least two courses that are basically dedicated for different techniques of how to integrate, because it is so much more complicated than differentiation. If it was the same, then they would just give you rules, and you would be done.
Now, the problem here is also that some techniques of integration (e.g. "u substitution") hinges on the idea that you understand how to differentiate (this is how you find your "du" to replace with "dx" or whatever).
You can learn integration at a very shallow, robotic level possibly, but so much of even basic integration uses derivatives that you are going to have trouble doing anything with it, or even getting it beyond memorization.
The thing about physics classes though, esp. at lower levels (maybe a mechanics or electricity and magnetism course), we do not challenge you with difficult integrals, just regular ones. The hard part about physics is the physics, not the math. Here is what you probably need to know
(1) how to integrate powers of x, e.g. x^n
(2) how to integrate trig functions (cos(x), sin(x), sec^2(x), tan(x), maybe sec(x) if the exam writer is feeling evil)
That should be it though. Study those, and consult with your professor/teacher! They can help advise you on what you need to know regarding integral calculus. It is best not to work blindly on your own, your teacher/professor is there to help.
