Techniques of Integration

Now that we are embarking into the world of multiple integrals it is essential that everyone has a good handle on basic techniques for solving single integrals.

Basic Integrals:
These are problems where the integrand is a "basic" function like $x^n$, $e^x$,$\frac{1}{1+x^2}$,  $\sin(x)$, or $\cos(x)$. There are several(1,2,3) places on the web to find these basic indefinite integrals. While I see no problem in using charts like these as a reference, I recommend not becoming too dependent on them.

u-substitution:
I like to separate these types of integrals into two types. The first is "baby u-substitution" these are integrals where the substitution can be done "in your head" without writing anything else down. Examples are as follows(click to see a solution):



The second type of u-substitution are slightly more complicated. Some examples are as follows(click to see a solution):



Integration by parts
There are two main type of integrals where the method of integration by parts is used. The first involve integrands which are the product of a polynomial and a tracendental function (like $e^x$ or $\sin x$). An example is as follows:

The second type are integrals involving inverse functions, like the following(click to see a solution).


Here is a pretty good write up on nearly everything about integration by parts.

I'll add to this page a bit later, but I want to go ahead and post it so you guys can see some examples of tricky integrals.