TITLE: Why do we need to learn integration techniques?
QUESTION [73 upvotes]: After a lifetime of approaching math the wrong way, I took two college math courses this quarter with a newfound zest for math.  These classes are integral calc and multivariable calc. 
Integral calc started out okay, learning about Riemann sums and the Fundamental Theorem of calculus.  But instead of spending a great deal of time gaining the intuition behind these things, we jumped into integration technique after integration technique.
Why on EARTH would we need to memorize and regurgitate a bunch of integration methods on toy problems for 6 weeks?  It's absolutely bizarre.  I'm not taking this at some random JC either, this is at a top research university.  This class is single-handedly destroying my enthusiasm for calculus.
When we have software that can do much more difficult integrals than we can with pencil and paper, why would we waste time memorizing Trig substitutions or integration by partial sums?  Doesn't that just make it a glorified algebra class?

REPLY [8 votes]: This is a fairly old question, but I didn't see an answer that reflected my point of view. So here's my two cents. There are three primary reasons (other than the ones that have been belabored already) that I believe learning integration techniques are worth-while:


Learning integration techniques reinforces the importance of duality.


The Fundamental Theorems are beautiful in part because they encapsulate the inverse relationship between differentiation and integration. It would be a shame to not see this duality carried as far as it could be. For example, we might expect that every property that differentiation satisfies, there might be an inverse relationship that integration satisfies. And, indeed, integration by parts can be thought of as the "opposite" of the product rule. And change of variables can be thought of as the "opposite" of the chain rule.
Pursuing duality whenever and wherever you find it is a good habit to form as you become a mathematician. Integration techniques are perhaps the first baby steps to developing this habit. However, this is perhaps the most superficial reason to appreciate integration techniques though.


Integration techniques are computationally useful for proofs.


More so than the exercises would have a student believe. It's true that most of the beginning problems in a calculus text are rather repetitive and tedious. These problems however are sort of like training wheels.
One of the best ways to exemplify that integration techniques are useful is to explore recurrence relations. These types of problems are usually some of the latter exercises in calculus texts. For example, if we defined
$$I_n=\int_0^\pi \sin^n x\, dx$$
for every natural number $n$, integration by parts would imply
$$I_n=\frac{n-1}{n}I(n-2)\,.$$
This small observation formed the basis of John Wallis's proof of the identity
$$\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdots=\frac{\pi}{2}$$
which is quite beautiful in itself, relating $\pi$ to the odd and even positive integers.
It should also be mentioned that one of the earliest and most common proofs that $\pi$ is irrational invokes integration by parts.
A similar technique with the logarithm could should that
$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots =\ln 2$$
which is also aesthetically interesting.


Leaning integration techniques informs you about what is possible and hints at what is impossible. These hints of impossibilities lead to some beautiful mathematics.


There is a quote that is relevant here. Namely,

"Common integration is only the memory of differentiation..." - Augustus de Morgan

Integration is interesting precisely because it is difficult. Without the FTC, calculating integrals with Riemann (or Upper/Lower) sums is tedious and frequently requires a spark ingenuity which changes dramatically between the functions you want to integrate. With the FTC however, many elementary functions become quite easy to integrate. With a powerhouse of a tool at our disposal it is natural to go on and use it to calculate the things we wonder about. Some of the problems that you encounter in calculus texts are sometimes actually problems that the author came across naturally themself or found novel enough to include.
With elementary functions mastered, it is natural to move on to more complicated functions to probe the limits of the FTC, integration by parts, change of variables, etc. to see where we eventually hit a wall. Finding the limits of your shiny new toy is also a good habit to form as you become a mathematician. The integration techniques you learn in a standard calculus course is a roadmap through the genius of past mathematicians that you don't have to repeat or would have difficulty redeveloping on your own.
Despite the swath of functions you're able to integrate with these techniques, it is interesting that there is a wall. Namely, we cannot integrate
$$e^{-x^2}\qquad \frac{\sin x}{x}\qquad\frac{1}{\ln x}$$
using the techniques that we learn from Calculus. We would not have been able to find these functions without getting our hands dirty through extended and purposeful effort. Few courses take integration to this extreme, much less prove that we cannot integrate these functions in terms of our other elementary functions.