Aside from this two practical technique to compute any integral, what else? Aside from this two practical technique to compute any integral, what else could called a fundamental method but not approximate method like Riemann Sum?
These two method I've been referring to are called integration by parts, integration by substitution(changing of variables)
 A: Some helpful links:
http://www.intmath.com/methods-integration/methods-integration-intro.php
http://tutorial.math.lamar.edu/Classes/CalcII/IntTechIntro.aspx
Paul's Online Notes are the best for learning calculus :)
A: I wouldn't call these two methods particularly fundamental. The way I see it, there are two basic ways to compute an integral


*

*Computing it directly from the definition of the integral, i.e. as some kind of limit.

*Using properties of the integral to  reduce the integral in question to one that is already known.
There are lots of ways to accomplish (2), and integration by parts and substitution are merely two well-known ones. Other that come to mind are


*

*Differentiating under the integral sign

*Swapping the integration order. For originally one-dimensional integrals, the first step is to somehow interpret them as a multi-dimensional one. In particular, this technique works to integrate $\int_{-\infty}^\infty e^{-x^{-2}}$.

*Contour integration. Here, one basically reduces an integral to a sum of of integrals of $\frac{c}{z}$ along closed paths around $0$.
But there surely are many others.
