How important is it to remember computational tricks as a pure mathematician?

Basically my question is whether the typical mathematician would consider it important to remember purely computational tricks like "trig substitution" which only help one solve obscure integrals that don't ever seem to be of interest in pure math. I can definitely understand things like integration by parts or even partial fractions but I took Calc II last semester and I have already forgotten most of the material. Can the typical mathematician pull things like this out of his rear or does one have to teach to remember things like that?

Talking about computational tricks more generally than just integrals, it wildly depends on the field of mathematics. Some have a lot of things that turn out to be fundamental that, to an outsider, seem like computational tricks more than deep theorems, or even just very useful tools. Most fields will have something. Just to name a few examples:

Computing continued fractions and rational approximations quickly is a very useful skill if you work in the right subfield of number theory, but is a curiosity to most mathematicians.

If you study fields related to Complex Analysis, especially from a more geometric view-point you'll develop massive skill at evaluating wonky integrals using clever contours.

If your a combinatorist, you might spend a lot of time calculating the values of summations that require arcane tricks and clever substitutions.

As has been mentioned in the comments, trig substitutions are secretly super important, especially things like $\tan(t/2)$.

So what about your situation? First of all, integration by parts is an exceptionally important theorem in general. I personally think they give people the wrong formula in school and that it should be written as

$$\int u\,dv +\int v\,du =uv\quad \text{a.k.a.}\quad u\frac{dv}{dx}+v\frac{du}{dx}=\frac{d}{dx}(uv)$$

because once I realized that integration by parts and the product rule were the same thing, remembering the formula and also figuring out how to do it in concrete instances became way better. Most mathematicians don't have a whole list of formulae memorized, but rather know how to derive them, because they understand the things that are hiding behind the curtain. So learn that. Learn how to think about the problems, and the particular tricks will be five minutes of scratch paper away whenever you need them.

This is rather a matter of preference. I know mathematicians that remember a lot of "tricks," but I also know mathematicians that do not care about memorizing "tricks" so much. Thus, answer to your question will largely depend on personal preference.

In my opinion, remembering basic and useful tricks comes in handy. However, sometimes it is hard to determine at the start what is useful and what is not. Also, of course it is impossible to remember every possible "trick" that you encountered.

If I were you, I would not worry that much about remembering tricks. Try to understand them first. If you understand the concept, re-learning the trick will be much easier. Also, there is a big chance that understanding the concept will help you to automatically remember the "tricks."