For u-substitution, what is the mathematical systematic way of picking a u? What is the systematic way of figuring out what u should be set equal to? 
In class we were introduced to u-substitution, which is the chain rule in reverse. You pick a u that is the inner function find the derivative, substitute to make solving integrals easier. My only problem is in the complex integrals he solved, it seemed as if he just randomly picked what he wanted u to equal and it magically worked. Is there a step I am missing? Maybe, it is really obvious to figure out what u should be, but I don't know the really obvious way to tell. What is the systematic way of figuring out what u should be set equal to? 
 A: $u$-substitution is basically the chain rule in reverse. 
Just like using the chain rule, you want to pick judiciously, and you'll learn how to do this by doing lots of problems and seeing some common patterns come up. 
A: The systematic approaches are nearly impossible to describe - it takes books and books and books, relying on mathematics well above standard calculus, especially when you consider that there are innumerable integrals that don't have solutions in terms of standard elementary functions.
There are several processes in mathematics where forward is easy but reverse is hard.  That, in fact, is what allows encryption on the web to work.  When you get into differential equations, often times the main point is to setup the problem, not solve it, or to use some method to come up with an estimated solution rather than an exact one.
The best way to be able to pick a "u" is to have your derivative table memorized, and be well-practiced with complex derivatives and what they look like.  This helps you "see" what you should pick for a "u".  If you apply the chain rule a lot, it is then much easier to see what it might look like in reverse.
But, as Batman said, it tends to be as much as an art as anything.
If you want to know more about how a computer might do it, you should pick up a book on computer algebra algorithms.  However, there are integrals that are solvable that are beyond the reach of humans, too.  As I said, forward is simple but reverse is hard.
