Integration By Parts
This chapter explains how to mentally do integrations which must be solved by integration by parts. As you read, it may seem that you have to keep an unreasonable amount of information in your head. ``How am I supposed to keep track of all this? It's not possible.'' What you are learning to do is work with your internal representation of the math. If you visualize an equation, that is one possible internal representation. The more efficient the representations you employ inside of you, and the more efficient your methods for working with them, the more complex the math you can easily do. One thing talented mathematicians have working for them is that they have discovered how to internally represent math expressions in ways that are both efficient and amendable to mathematical manipulation. This lets them use the mental energy they have very, very effectively. As mentioned earlier, if a person's ability to visualize is good enough, they can just imagine a whiteboard in their mind's eye, and imagine each step written out exactly as they would write it out on a real whiteboard. That's certainly one possible internal representation, and one that requires a LOT of mental work! Mathematicians learn to represent expressions in more minimal ways, making them much easier to manage. That is precisely what you will learn to do here!
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