 All right, how about another anti-derivative? How about e to the 5x? So we want a function whose derivative is e to the 5x, and again, we might recognize this. We should recognize this as an e to the function. And we know that the derivative of an e to the function is the same thing and some other stuff. So we might start out with e to the 5x and see what happens when we differentiate it. So there's my e to the 5x. Derivative is world's easiest derivative. The same thing, e to the 5x, times derivative of what our power is. Now, again, what I want is an e to the 5x. What I have is a 5 e to the 5x. So I can multiply by one-fifth to eliminate that constant. Over on the left-hand side, I have constant times derivative of function. So I can move that constant in, and my derivative one-fifth e to the power 5x is the same as e to the 5x. And so my anti-derivative is that function, plus any constant that I care to name. And maybe I'll name this constant George. All right, how about the anti-derivative of x times x squared plus 3? So let's do a little bit of analysis. This function that we're looking at is a product of two things, and the derivative of a product gives us a product. And one of the things that we might remember from the product rule is we always end up with echoes of the original function. So what we want is some function that's going to include x squared plus 3 and also x. So what's the simplest thing we could do? Well, we could just look at the derivative of that product. And we do have to apply the product rule combined with, don't forget, the chain rule. And we end up with that as our derivative. And two things happen, both of which are bad. One is that there's no constant I can multiply the right-hand side to get my original function x times x squared plus 3. And what that means is that I can't multiply this by a constant to get this, which means that what we've been doing to find the anti-derivative won't work. So what can we do? Now we can always do algebra. So let's take a look at that function again. That's x times x squared plus 3. That's the same as x cubed plus 3x. And I've gone from something that's a product to something that is a sum. And that's good because finding the anti-derivative of a sum is a lot easier. So I can find a function whose derivative is x cubed, a function whose derivative is 3x, and add the two together. So there's my function whose derivative is x cubed. And I want a function whose derivative is 3x. So I'll start with x squared. The derivative is close but not quite. And I'll multiply by 3 halves to get what I want and move the constant inside. And so the derivative of a sum is the sum of the derivatives. So that tells me the anti-derivative is going to be the sum of a quarter x to the fourth, three halves x squared, and then some constant.