First of all, there's a small mistake in your work leading to your first displayed line. You'll want to fix that first. Now, you say you want to integrate both sides. To convince yourself this is a valid step, think of it this way: you have an expression of the form Function1 = Function2. We have the theorem:

Two functions are equal if and only if their antiderivatives differ at most by a constant (provided, of course, that they permit antiderivatives).

You ask if your limits of integration are correct. Just do this: verify that the integral represents SOME anti-derivative of the original function. (When you take its derivative, do you get back to the original function?) Then the theorem applies. At least, that's how I understand it. I, like you, try to also be explicit about which theorems I'm using for problems like these. That's good. People can be careless about using the FTC.