 Let's look at some examples of computing anti-derivatives. Now anti-derivatives is reversing the derivative process. When we have a derivative problem, it's given us a function, we have to find its derivative. Anti-derivatives mean we are given the derivative, what's the function? And every derivative rule that we've learned in the past actually can be turned into an anti-derivative rule. One to start off with is with the power rule. We've learned with the power rule that if you take the derivative of a power function, we can find its derivative by bringing down the exponent as a coefficient and we can lower the exponent by one. So when we take like d dx of a power function x to the n, the idea is you can bring the exponent down and so you end up with n times x and then the exponent lowers by one as well. So you get n times x, n minus one there. That's the derivative of power function. Well, if we reverse this process, we would increase the exponent by one. So we wanna increase the exponent by one and then bring this thing back inside. That is we were multiplying by the exponent and we subtract one from the exponent and we reverse this process. We're gonna add one to the exponent like you see right here and then we divide by that new exponent n plus one. So this gives us what we might call the power rule but this is the power rule for anti-derivatives or we might call it the anti-power rule, right? It's like the nemesis to the power rule. It's the kryptonite for the power rule. Now the nice thing about anti-derivatives is because we're good at calculating derivatives, we can actually, if we ever wanna check our answer for an anti-derivative, we just take the derivative like we did earlier in this class here. So I claim that the anti-power rule says that the integral of x to the n will be x to the n plus one divided by n plus one plus a constant. And to check this, I'm just gonna take the derivative of what I think the anti-derivative is. So we take x to the n plus one over n plus one plus a constant or take the derivative. When we take the derivative of a constant that'll just vanish and then by the power rule, we're gonna bring the exponent down n plus one. Well, I guess actually let me go back a second. Constant multiples can be brought out of the derivative process. So we get x to the n plus one. We have to take its derivative. The one over n plus one we brought out. And so now by the power rule, we will bring down the power of n plus one and then the exponent gets lowered by one. So n plus one minus one. And as we simplify this, n plus one divided by n plus one just goes to one and plus one minus one just goes away as well. And we're left with just x to the n, which shows us this anti-power rule in fact gives us the correct anti-derivative. Let's look at some examples of this. So by the power rule, if we take the integral of t cubed, the anti-power rule says we'll increase the power by one. So three plus one there. And then we divide by that new power, three plus one. Don't forget your arbitrary constant there. You have to add a constant. And so if we simplify this thing, you get t the fourth over four plus a constant. This is our anti-derivative. Sometimes if you want, you can write it as a coefficient, one fourth t the fourth plus a constant. That's really just giving our function a manicure at that point. It doesn't change who it is on the inside. Exact same function. And if you're all concerned if this is the right anti-derivative, just take the derivative again. One fourth t the fourth plus a constant. Take the derivative. Well, you're going to get by the power rule four over four t cubed plus zero, which simplifies to be t cubed. We see we have the right anti-derivative. Well, how about the next one? Well, much like we did when we first learned about derivatives in order to apply the power rule, we have to view this as a power function. So if we take the integral of t to the negative two, that's a power function. One over t squared is just t to the negative two. Reciprocals mean negative powers. Now by the power rule for anti-derivatives, we're going to increase the exponent by one. So we add one to it and then we divide by this new exponent. And don't forget the plus c. Otherwise you'll probably get docked by your math professor there. If we increase the power by one, we get t to the negative one. This will sit above now negative one plus a constant. Let our function go to the day spa. We end up with negative one over t plus an arbitrary constant. And this is our anti-derivative. You can take the derivative of that again to see that this was the correct anti-derivative. As another option here, another example, take the square root of u here as our integrand. First recognizing as a power function, we can write this as the one half power. Take the integral with respect to u here. So by the anti-power rule, we'll increase the exponent by one and then divide by that new exponent. Don't forget your constant here. And so this is the hardest part of the whole lecture. Oh no, what are we going to do with a fraction? Well, one's the same over as two over two. So we're going to take one half plus two halves here, which gives us three halves. And then we're dividing by three halves at a constant. Now, whenever you divide by a fraction, remember you're just multiplying by the reciprocal. And so you end up with two thirds u to the three halves and don't forget that constant. We'll see later in this lecture why we care about the plus c so much, but it is because we don't know what that constant is. We have to include it because it could be anything, honestly. We'll talk about this in another part of this lecture, not in this video though. Now, for this last example right here, it's like, okay, let's take the anti-derivative of there's no function. Where's the function? Well, we saw this same situation when we dealt with derivatives. There is a function. It's just the function is one. You don't see it there because multiplication by one is implied. And okay, what do you do with one? It's a constant. Well, this is still a power function because one is just x to the zero dx. And so by the power rule, you're going to increase the power by one, which gives you one and then you divide by one. So we end up with just x to the first over one plus a c. Make it look like a one so you don't accidentally take the derivative again. And so the function you're looking for here is x plus c. All right? And you can double check again that the derivative of x plus c is just one. So this is in fact the correct anti-derivative. Now, there's one thing I almost forgot to mention. When it comes to the anti-power rule, you can apply this for any power function except, except when n is equal to negative one. The power rule doesn't apply there because if you try to do it, if you have x to the negative one dx like so, well, the power rule would say, okay, I'm going to take negative one plus one and divide by negative one plus one plus a constant. This gives you x to the zero over zero plus a constant. And uh-oh, uh-oh, we divided by zero right here. This of course represents death. You don't want that to happen, right? So this is our skull and crossbones here. Oh boy, it's bad. We don't want to divide by zero. But actually when it comes to negative one, we have to handle this thing a little bit differently. If you want to take the integral of one over x dx, well, let me back up for a little bit. If you want to take the integral of x and negative one dx, I actually recommend we think of it as the function one over x dx instead. The reciprocal, because then we have to find a function whose derivative is one over x. And we actually know that function, it's the natural log of x plus a constant. And that's almost correct, right? The issue here is that the domain of one over x is all numbers except for zero. The domain of the natural log is only positive numbers. And as a function's derivative never has a bigger domain than the original function, we actually have to make a quite a simple modification here. The anti-derivative of one over x, x and negative one power, this is gonna be the natural log of the absolute value of x plus a constant. The absolute value allows that the domain of both of these functions is gonna be everything except for zero, all right? And so when you can use this power function for any anti-derivative power functions, this power rule, except for when you have the function one over x, use the natural log instead.