TITLE: What's the deal with integration?
QUESTION [9 upvotes]: So at uni we learned tricks and techniques for integration until cows came home.
But to what end? Any/All definite integrals can be evaluated using numerical methods. Most integrals in application can not even be evaluated in elementary terms anyway.
So is integration like calligraphy?, where it is pretty to do/look at but a printer is the practical way to go.
Or are there other reasons behind learning evaluation of integrals in elementary terms beyond grading or personal joy?
PS : By practical application we don't mean oversimplified situations. Even the equation of pendulum is simplified so a nice answer for simple cases can be derived. In reality the change of potential energy with distance like calculations effecting satellite and probe launches to other planets can only be evaluated using numerical methods.

REPLY [2 votes]: Most integrals in application can not even be evaluated in elementary terms anyway.

This statement should be considered carefully and not taken as a general truth. Everything depends on what you mean by "integrals in application." 
Yes, integrals that arise in scientific/engineering practice are typically complicated and require numerical methods to evaluate.
On the other hand, one of the most important reasons for one to learn integration is to understand one of the mathematical pillars supporting the entire theory of physics from the 17th century onward. If you have ever taken a serious physics course (classical mechanics, for example) then you do see exactly solvable integrals in the derivation of many formulas or approximations. Students who do not know the basics of integration cannot progress very far in physics because it is absolutely necessary to understand the reason certain formulas should hold.
Of course, this is not at all unique to physics. Even though in practice engineers almost never compute an integral by hand, when they are learning their trade they must learn to do so because integration is a technique that is required to understand the theory behind the material in their engineering classes.
To summarize by analogy, integration techniques are like learning the alphabet. You need to do it to read the language of the sciences (e.g. engineering and physics), but once you do know how to read you can get by without remembering what letter comes after Q.