Are there parts of Integral Calculus that just *have* to be memorized?

I'd say that it is more a question of pattern recognition than anything. You may have to memorise a little bit to get going, but after then you should look at the integral and get an idea of whether you need a spanner or a screwdriver or a wrench.

I have long since forgotten what the integral of $\frac{1}{\sqrt{1-x^2}}$ is, but I remember that it looks $x=\sin\theta$-ish, and I try that. $\frac{x}{\sqrt{1-x^2}}$ has more of a $y=x^2$, $dy=2x\ dx$ feeling to it, though I may be wrong.

I've never memorised MathematicsStudent1122's $\int (\sin x)^n \ dx$, but I dare say that if I found myself having to confront it daily, I would remember it after the first few workings out.

Don't rot your brain with a calculator, because it will teach you nothing. People aren't asking you to work out integrals because they want to know the answer! If all else fails, work out a Taylor series, integrate term by term, and see if the answer looks familiar.