 We've been learning how to differentiate all semester, now we want to go backwards. So let's throw down a definition of an anti-derivative. Suppose the derivative of capital F is equal to lowercase f of x. Then capital F of x is called an anti-derivative of f of x, and we use the notation squiggle f of x dx equals capital F of x, where the squiggle is our symbol for anti-derivative. We say that f of x is the integrand, and dx is the differential. And here's a very, very, very, very, very, very, very, very important idea to keep in mind. The differential variable should be the only variable in the integrand. If there are other variables in the integrand, we can't proceed, at least not until we take calculus 3. So for example, we want to show that cosine squared x plus 5 is an anti-derivative of minus 2 cosine x sine of x, and then find two other anti-derivatives. So we'll pull in our definition of anti-derivative. It's an anti-derivative if the derivative is what our function is. So the easiest way to verify this is to differentiate. Is it true that the derivative of cosine squared of x plus 5 is 2 cosine x sine of x? Well, let's check it out. So I want to find the derivative of cosine squared of x plus 5. And since this is a sum, we know the derivative of a sum is the sum of the derivatives. Now, remember that cosine squared of x is shorthand for cosine of x squared. And so this function is a squared function. It is a something squared. And the derivative will be 2 something to the first times derivative of our something. Meanwhile, here we want to find the derivative of 5, which will be 0. And we have an unresolved derivative here, derivative of cosine x. And the derivative of cosine x will be minus sine of x. And so our derivative will be minus 2 cosine x sine of x, which is what we want it to be. How can we find other anti-derivatives? The important thing to recognize is that in our derivative, we took the derivative of a constant, which was 0. And that's going to be true for any constant at all. And so changing the added constant won't change the derivative. So instead of cosine squared of x plus 5, we could have cosine squared of x plus 10, or we can show off and say cosine squared of x plus this horrible mess. And each and every one of these has a derivative equal to minus 2 cosine x sine x. And this leads to the following theorem. Suppose f is an anti-derivative. Then f of x plus c, where c is any constant at all, is also going to be an anti-derivative. How do we find anti-derivatives? So suppose I want to find the anti-derivative of 5x cubed dx. What this means is we want to find a function whose derivative is 5x cubed. And before we proceed, one important check to make. Our differential variable is x, and our function only has variable x in it. And so it's possible to proceed from this point in a meaningful way. But how? We'll fall back on a time-honored strategy in mathematics and in life. We can proceed by guess and fix. We'll guess an answer, and if it's correct, great. If not, we can fix the answer so that it'll be at least more correct. Remember, you don't have to be right at first, as long as you're right at last. So what we want is something whose derivative will give us 5x cubed. And it's helpful to remember a couple of basic derivative rules. The derivative of a constant produces zero. The derivative of a sum produces a sum of derivatives. And the derivative of a constant multiple of a function produces a constant multiple of a function. And this last rule is fairly important, because it means that we can ignore the constant, at least until the very end of the problem. So let's focus on this x cubed. What sort of thing do we differentiate to get an x cubed? Well, one of the very first derivatives we found is that if I have a function x to the n, then the derivative will be nx to the power n minus 1. And so that means if I find the antiderivative, I can increase the power. So this x to the third might have come from the derivative of x to the fourth. Well, let's drop that in and see what happens. If my original function was x to the fourth, then the derivative will be 4x to the third, which is exactly the same as what I want. Oh wait, no, it's not actually exactly the same as what I want. So let's fix it. I don't want a 4 here, I want a 5 there. So let's multiply by 1 fourth, that'll take care of the 4, and then multiply by 5, which we'll put in the 5. And because these are constants, I can multiply them together. And because they are constants, I can move them into the differentiation. And it's useful to remember one more important thing. Once I find an antiderivative, then I can form other antiderivatives by adding any constant at all. And so the antiderivative isn't just 5 fourth x to the fourth, but it's actually 5 fourth x to the fourth, plus any constant that you want. Let's try another antiderivative. So we want to find a function whose derivative is e to the 3x. So we might ask ourselves, self, what produces something that looks like e to the 3x? And we might remember the world's easiest derivative. The derivative of e to the x is e to the x. And so maybe this e to the 3x came from e to the 3x. Well, let's check it out. If I differentiate e to the 3x, then I get e to the 3x times 3, because we have to use the chain rule. And it's not quite what we want, but that's okay. We can fix it by multiplying by 1 third. And again, any added constant won't change the derivative. So our antiderivative looks like it's going to be 1 third e to the 3x plus c. That was fun. How about another antiderivative? As usual, it's helpful to identify what type of function we're dealing with here. In this case, our integrand is a sum, and we know that the derivative of a sum is the sum of the derivatives. So this 5x cubed plus e to the 3x must have come from a sum of derivatives. So we need to find something whose derivative is 5x cubed, which we already did, and we need to find something whose derivative is e to the 3x, which we already did. Now, note that we didn't include the constants. What we can do is we can lump all of those constants together into a single constant right here at the beginning. All right, let's look at another antiderivative. This is a product. And unfortunately, none of our derivative rules give us a product. Remember, even the product rule gives us a sum. So we should try something else. We could always try some algebra. This is a product, and we can multiply out. And so instead of trying to find the antiderivative of a product, which we have no rules to handle, we can instead try to find the antiderivative of 6x squared minus 13x minus 5. So we'll invoke gas and fix. We want to find something whose derivative is 6x squared. So again, I'm guessing this is going to come from something like x to the third, except when I differentiate x to the third, I get 3x squared, which isn't what I want. But if I multiply it by 2, I will get what I want. And that factor of 2 becomes part of the antiderivative. I also want to get a 13x. That's 13x to the first. And so that probably came from something like x squared. But when I differentiate x squared, I get 2x. So I need to fix it. And I can include that factor of 13 halves as part of the antiderivative. And then finally, I want that antiderivative of 5. And so that must have come from something like 5x. So I can put the pieces of the antiderivative together and don't forget, plus any constant at all.