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 relationships between differentiation and integration. It would be a shame to not see these inverse relations carried as far as they 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. This is the most superficial reason to appreciate integration techniques though.
- Integration techniques are actually computationally useful for proofs.
More so than the exercises would have a student believe. Some of the best ways to exemplify this is the discovery of recurrence relations. 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.
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 are two quotes that are relevant here. Namely,
"Common integration is only the memory of differentiation..." - Augustus de Morgan
"It can be of no practical use to know that $\pi$ is irrational, but if we can know, it surely would be intolerable not to know." - Edward Charles Titschmarsh
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 these 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. The techniques you learn is a roadmap through the genius of past mathematicians that you don't have to repeat, but 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.
It should also be mentioned that integration by parts also leads to an elementary proof of the irrationality of $\pi$...