 There are only two integration techniques, u-substitutions, and integration by parts. Everything else is algebra, and so, a key strategy, try the easy things first, and so, of our integration techniques, u-substitutions are easier. So, for example, let's evaluate this integral. Now, this is usually done using trigonometric substitutions, but remember, a problem exists whether or not you know how to solve it. So if you don't know trigonometric substitutions, you still have to evaluate the integral. And even if you do know trigonometric substitution, remember, if the only tool you have is a hammer, you have to treat every problem as a nail. And that's okay if you're trying to hang pictures. That's so good if you're trying to open a jar of pickles. So the key idea behind u-substitutions is to sweep it under the rug. And in this case, since the radicand is the messy part, let's try u equal the radicand. And so, du will be... Now, our differential requires us to have a minus 2x dx, and remember, you can have anything you want as long as you pay for it. So we'll put in the minus 2x dx, and we'll pay for that by splitting off one factor of x from the x cubed, leaving us with an x squared, and the negative 2 we'll pay for using a factor of negative 1 half, which we'll put out front of the integral. Now, remember, the differential variable is the only one allowed. So we have to do something with this x squared, and our substitution allows us to solve for x squared as... and so we can replace. If we're not sure about what to do with our next step, it's useful to remember, if in doubt, expand or factor. This is already written as a product, so we can't factor it any further, but let's expand it out. And now we have a difference. So remember the integral of a sum or difference is the sum or difference of the individual pieces, and so we get... and rewriting using our exponential form and integrating. And remember, put things back where you found them. So u was 1 minus x squared, so we'll replace. And it's also useful to remember, we should answer questions in the same language they were asked. While there's nothing wrong with this answer, because our integrand was originally given to us in radical form, we should probably convert these fractional exponents into radical expressions. And so our final answer in its best form would look like...