 Now there's a number of ways of finding antiderivatives, but one of the more useful is known as a u-substitution. We can find the derivatives of more complicated expressions by recalling the chain rule. For convenience, it helps to think of making a u-substitution and pretending that du dx is a fraction. And we'll proceed as follows. Let u be some variable expression, find du dx, solve for dx, and replace in our antiderivative f of x dx. And here's the important check. If the resulting expression is only in terms of u, we might be able to evaluate it. Now finding the best substitution to make is an art. No formula or algorithm will tell you what will work. However, there are some useful guidelines. First, when making a u-substitution, never let u be a constant. Next, never let u be x. On the other hand, it's helpful to choose u to be part of an expression, exponents, denominators, radicands, arguments, or factors. And finally, if your first substitution doesn't work, try a different u-substitution. There are other things we can do if a u-substitution doesn't work, but they're a lot more complicated, so we'll want to try the simple things first. For example, suppose I want to find the antiderivative of e raised to power 3x. I can let u equals e. Well, no, I can't. We should never let u be a constant. E is not a variable expression, so it's not going to be useful as a substitution. I could let u equal e to the x, but this is the whole function. So again, this is not going to be useful. We want to let u be part of an expression, so let's try u equals 3x. So if I let u equals 3x, then my derivative will be 3, and pretending that du dx is a fraction, I can solve for dx. And so I find dx is 1 third du. I'll substitute that into my original antiderivative, and because 1 third is a constant multiplier, I can move that to the front. And now I have to find the antiderivative of e to the u, which is the world's simplest antiderivative. It's just going to be e to the u. Plus, don't forget, our constant. And finally, we apply the kindergarten rule, put everything back where we found it. And in this case, I substituted u equals 3x. I should put that back. Now put another antiderivative. So one possibility is to let u equal sine of x. So I differentiate. I pretend du over dx is a fraction and solve for dx. I substitute. And after all the dust clears, I have the antiderivative of u, which is going to be 1 half u squared plus c. And I'll put everything back where we found it. And there's our antiderivative.