Why do we need to learn integration techniques?

This is, in my opinion, a common feeling after a "lifetime of approaching math the wrong way". People are taught math in a very rigid rule-based formula/pattern method, and then when they contrast this against mathematical proofs they have a knee-jerk reaction against anything which looks even remotely like what they did before. The fact of the matter is, however, that you will need to be able to do some of this without aid of a computer.

When reading a proof, it is easy to take for granted that you are able to fill in the details between steps of the proof, when really all these steps are able to be filled in precisely because you have the understanding of solving equations (basic algebra) and working with inequalities (basic arithmetic), for example. The same thing holds true for proofs involving integration and derivatives.

Perhaps it's best to leave it to those who really know what they're talking about - Spivak writes in his chapter on integration that our motivation should be that:

1. Integration is a standard topic in calculus, and everyone should know about it.
2. Every once in a while you might actually need to evaluate an integral, under conditions which do not allow you to consult any of the standard integral tables.
3. The most useful "methods" of integration are actually very important theorems (that apply to all functions, not just elementary ones).

He emphasizes that the last reason is the most crucial.

I would personally advocate that students should be wary of falling into the trap of thinking that such pedantic methods are beneath them. It is often easy to think you understand something at a high level, but you don't truly learn what it is all about until you really get your hands dirty with it.

A short answer from Isaac Asimov : 'The Feeling of Power'.

Here is a little fable. Some parts may resemble actual events that occurred someplace or other. Other parts are purely made up.

A large university offers different types of calculus courses for the benefit of various other departments. One day the Mechanical Engineering curriculum committee comes to the Mathematics curriculum committee.

"Look!", they say. "If we tell prospective students that we require $X$ semester hours of math for our major, but a competing university says they require only $X-2$ hours, that other place will win the students. So we have to cut $2$ hours from the required math courses."

"OK," Math says. "What do you want to cut?"

Mechanical Engineering studies the syllabi, and says (among other things), "Cut out techniques of integration. Students can do all that by computer, anyway."

So the new--sleeker--course for Mechanical Engineers comes to be.

A few years pass.

Big important Senior Professor is teaching a course for Mechanical Engineers. He derives an equation for this problem he is doing. Then he says, "Now we integrate by parts to put the problem in this other form, so we see it means that we should minimize the energy."

The students reply: "Integration by parts? We've never heard of that!" (Actually, integration by parts had been mentioned a couple of times in their math course, but they had no baby problems to practice it on, and thus they have forgotten it.)

NOW Mechanical Engineering is accusing Mathematics of shoddy work... The members of both curriculum committees have changed by then, so neither department is likely to remember the reason that techniques of integration was omitted from the integral calculus course.