 Alright, let's take a look at another example of a U-substitution. So the most important thing to remember is, if the first thing you try doesn't work, give up. Throw your hands in the air and say, I can't solve this problem. Well, actually the thing to remember is that try different things. There's many possibilities for a U-substitution. You've got to try them all. And unless you are phenomenally intuitive, the first thing you try probably isn't going to work. So a little bit of analysis goes a long way. Let's take a look at this. This is the antiderivative cosine x sine squared x dx. And a little bit of analysis goes a long way. This is a product. So we might try U equals one of the factors in the product. So U equals cosine x. Let's see what happens if we do that. So I'll try it out. Let U equal cosine x. Differentiate derivative is minus sine x. And again, du dx is not a fraction. But we can pretend it is one and try and solve for dx and make our substitution. So there's my original antiderivative of cosine x sine squared x and cosine x. Well, that's U equals cosine x. So I'll substitute that dx du over minus sine x. I'll make that substitution. I'll do some phenomenally difficult algebra. And I run into a problem, which is that the differential here defines the only allowed variable. U is the only variable I'm allowed to have. Well, that's good. But then I still have this x left over. And what that says is I cannot evaluate this antiderivative at this point. Well, I'll try something else. So this won't work as a substitution. I'll try something else. Now remember that we identified that we're trying to find the antiderivative of a product, cosine x, times sine squared of x. So our first attempt was to let U be one of the factors, cosine x. Now this U substitution didn't work, so we have to think of something else. Well, in a fit of brilliant inspiration, if U equals one of the factors didn't work, let's try U equal the other factor. So we'll try U equals sine squared of x, and I'll go through the same process. So I'll find the derivative. Remember, chain rule, chain rule, chain rule. So the derivative is going to be 2 sine x cosine x. And again, du dx is not a fraction, but I could still solve for dx as if it were. dx equals du over 2 sine x cosine x, and I'll make my substitutions. So again, here's my antiderivative, U equals sine squared of x. So that's gone dx du over 2 sine x cosine x, and a lot of horrendously difficult algebra, cosine x, cancels. I get this. And once again, I'm in the position where the only allowable variable is U. That's good, but the sine of x is still there, and so I don't have a proper substitution. I can't do this antiderivative. Well, it's like in baseball, two strikes and you're out. Well, actually, it's three, isn't it? We try things at least three times. So let's do a little bit more analysis here. This substitution, U equals sine squared of x. A little bit of analysis goes a long way. This function is really something squared. The last thing you do tells you the type of function. This is a something squared type of function. So maybe a better substitution is U equals something. And we apply the kindergarten rule, put everything back where you found it. So the thing that's being squared is sine. So maybe the substitution I want to try is U equals sine of x. I'll try it out. Let U equals sine of x. Do dx is cosine x. Solving for dx gives me dx equals du over cosine x. And so making that substitution, again, here's my original antiderivative. Cosine x sine squared x dx. U equals sine of x. So this part becomes U squared. dx is du over cosine x. Some horrendously difficult algebra. Cosine x cancels. And I get this. U is the only allowable variable. U is the only variable I see here. So it looks like we're good. And I can anti-differentiate. And again, kindergarten rule, put everything back where you found it. U equals sine of x. Replace the U with sine of x. And there's my final answer. Except again, you do want to check to make sure that the derivative of this is in fact what we started with.