Note : In this question I speak more from a calculation/operational point of view, as opposed to a more theoretical (Analysis) point of view.
When studying Differential Calculus, I found that there was very little that I had to memorize. Virtually all calculation aspects, such as finding derivatives etc., and some theorems, could all be derived on the spot through basic methods.
As examples, through basic implicit differentiation, one could prove the inverse function theorem, within a few lines.
$$\text{Inverse Function Theorem}\ \ \ \ (f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$$
Or if I wanted to find $\dfrac{d}{dx}\ \tan^{-1}(x)$, I could use the inverse function theorem and with the help of a trigonometric identity find the derivative quite easily. I didn't have to memorize $\dfrac{d}{dx}\ \tan^{-1}(x) = \dfrac{1}{1+x^2}$. In fact, apart from the derivatives of $\sin(x)$, $\sinh(x)$ and $\cos(x)$, $\cosh(x)$, I didn't memorize any of the other derivatives for trigonometric functions, I would just re-derive them using basic differentiation rules each time.
However I noticed that when studying Integral Calculus, there tends to be a lot more that one just has to commit to memory. For example if I wanted to evaluate the following integral $$\int \dfrac{1}{1+x^2}\ dx$$
The only way I could ever evaluate the integral, would be if I knew $\dfrac{d}{dx}\ \tan^{-1}(x) = \dfrac{1}{1+x^2}$, which would require that I had memorized the derivative (something I tried my best not to do when studying differential calculus).
When studying Mathematics, for the most part (and within reason of course) I try my best never to memorize what I can re-derive/prove. I've found that this approach helps improve my skills, and pushes me to search for the deepest possible understanding.
But it seems that there are some things, that just have to be committed to memory to be able to make any sort of progress, and this troubles me quite a bit, as I'm not sure as to what I should be just memorizing, and what I should really be working to get the best understanding on.
Furthermore Integration is a very heuristic process, whereas Differentiation is a more algorithmic process. Generally we try to get integrals into forms we know of already so that we can evaluate them (with the exception of the Risch algorithm), or it would be impossible to evaluate them by any other means. Wouldn't that require one to memorize the various types of possible integrals?
First off, am I looking at this wrong? Are there ways one can reprove results, or evaluate integrals, in a manner that doesn't require one to just memorize and recall a list of formula's like a parrot?
What aspects of Integral Calculus would you say, just have to be memorized, i.e. what results in Integral Calculus are close to impossible to re-derive or prove on the spot?
Where does one draw the line, between what should be looked at long and hard for the deepest possible understanding, and what should just be memorized?
Lastly, correct me if I'm wrong, but as one makes the transition into higher mathematics (analysis and beyond), that there are some things that you just have to commit to memory, to be able to make any sort of progress?
Best Answer
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.