So hi everyone, let's talk about antiderivatives some more. In particular, we're going to talk about indefinite integrals and this kind of gets back to a little bit about the first part of the fundamental theorem of calculus. So we're going to have some notation that comes into play here, and we're going to draw the integral sign with a function F of X DX. Now this is important. What is this thing actually mean? I'm going to put it in words first, and then I'll put in symbols, but this is equal to all the antiderivatives, all the anti derivatives of the function F of X. Okay? So all of them. Now this is weird, so if I ask you is this thing a number? No, not not at all. Is it a function? It's not a function, it's in fact lots of functions. So normally what you see is this thing is equal to the family of functions F of X + C. So just keep in mind this is not one particular function and it's very different than the definite integral, which is a number, so I'll put that in red. So the definite integral, by the way so we call this an indefinite integral. So let's put versus the definite integral which remember had bounds on it, so they look very similar. Remember, this is a number one is defined, so I have one of them that means infinitely many functions, antiderivatives of a given function, and I have another one that means the indefinite integral rule, which is a family of functions and versa number. So keep these things in mind and the notation between them is very subtle. So, what I'd like to do now is sort of get to a big table going of things that we know which we're going to use all the time and they come from our knowledge of differentiation. So for example, if I want to talk about X to a power, what is the anti derivative of X of DX? Again, so you have to kind of undo the power rule for differentiation and you can check X to the N + 1 / N + 1 and then because it's a family of curves plus see. Now how do I know this is true? How do I know that X to the N + 1 / N + 1 is the Anti Derivative X of the N? Well, you differentiate it. Bring the power down, subtract one. You get exactly this and now this is important to realize, this only is defined for N not equal to minus 1. The- 1 case minus one case, I guess we could talk about that. If I have X to the minus one, probably better known or more commonly written as 1 / X DX. What function has derivative 1 / X? So this would be the LN of X plus C, now for bookkeeping reasons, for domain issues, you have to put the absolute values in here. So this watch out for that little absolute value is a common place where people get stuck. So there's the two powers, so keep those in mind. Other things we know from just differential calculus, if I wanted an anti derivative of sine, what function has derivative sin of X? That's a minus cosine of X + C. We can also do cosine. What is the anti derivative? What is the definite integral? Indefinite integral of cosine X with function has cosine X sin of X + C. Let's keep going. What about power function here? So if I had A to the X DX what function has as its derivative ADX? Will be remember what the derivative here. Normally, if you're taking derivative of X, you multiply by the natural log of the base. So now we will divide by the natural log in the base. Now the fun doesn't stop there. There's lots of functions we know because we are masters of differential calculus. What function has the derivative secant squared? Remember, tangent of X + C. What function, let's do secant tangent's going to our trig functions here. What function has derivative secant tangent? That's good old secant. Plus see what function has derivative cosecant squared? What is the indefinite integral of cosecant squared minus cotangent? Keep going with other ones I'm missing here, so we can do what is cosecant X times cotangent X, I want the indefinite integral. So that's minus cosecant X + C. Let's put a couple more on the board here. How about we did, this one comes up a bunch of the anti derivative of 1 / X 2^ + 1 DX. What's that? That's Arctan, Arctan plus C. Let's get its buddy in there, how about 1 over the square root of 1- x squared dx. And that's arcsine, arcsine of x + C. And again, how do I know these are true? Take the right side, that's an equation, so you can rewrite the left, take the right side and differentiate the answer. Differentiate this thing, so practice your derivatives, and you will get back the function that is on the left. So this is our table of indefinite integrals, these are just sort of the ones we know from calc one, we know from knowing derivatives, that gets us these functions. Now, there are of course more, and I'm going to give you these just to kind of put them together, because they come up so much that I want you to know them. If I have a constant, if I have a constant k dx, so this is a particular case of the power rule. Just remember, this is k x + C, it's a particular k. So again, how do I know that? Take the derivative with respect to x and get back k. other things that come up are slight variations of the functions we saw. So we had cosine of x over there, but what if I have cosine of some constant times x? In that case, it becomes 1 over the constant, sine of k x. You can check that by using the chain rule to take a derivative of sine of k x, you'll get back exactly. And it's similar also for sine and cosine. So if I did the same thing for sine of k x dx, you're get back -1 over k cosine of k x + C. One more that comes up a bunch is the exponential. Obviously if you're doing modeling, population problems. If I have the antiderivative of e to the m x + b dx, what's the antiderivative, what is the indefinite integral? We have 1 over m e to the m x + b + C. So I want to put these down as well, and again, you should be able to verify all of these. Take the derivative of the right side, use the chain rule, and you will get the function in there as well, okay? So these are sort of the ones that we're taking with us. There are still lots and lots and lots of functions whose antiderivatives we need to figure out, but we're not there yet, we'll do that in another video. But let's, with our tables, with our tools in mind here, let's go do some examples that are a little tricky, or just a little clever, that are not immediately just spit back from the tables. So let's use the tables to do something. So I'm going to put an example on here, and I want you guys to think about it for a moment. Pause the video if you have to, if you can, and think about it. So here we go, I want the indefinite integral of sine of 2x over sine of x dx. Find the indefinite integral, pause the video if you can and work this out. This particular example is not on our table, but I claim you can still do it, all right, you ready? As always, when it's not on the table, it's going to need a rewrite. So there is a trig identity, sine of 2x, double-angle identity. So what does that become, it becomes 2 sine of x cosine of x. Sine of 2x is 2 sine x cosine, one of those trig identities. And if you rewrite it this way, again, as you rewrite it, don't forget the equals sign, don't forget the integral sign, don't forget the dx, the sines cancel. And therefore, you get the integral of 2 cosine x dx. Remember, constants, by the algebraic properties of the indefinite integral, they pop out. So you get 2, and then all I want back is the indefinite integral of cosine of x. That is sine of x + C, so put it all together, we get 2 sine x + C. All right, so not immediately obvious, but that was a good one. And let me do one more, same thing. Where I'm going to hand you a problem that is not immediately on the table, but I claim you can do it. And trig's always a good one for these, because there's just so many identities. So pause the video and try to work this out before moving on, ready? Integral of sine x over 1, here we go. So 1- sine x, I hope you recognize this. Sine of x, this is cosine squared, this is cosine squared of x. That's the Pythagorean theorem, that says sine x + cosine squared, sine plus cosine is 1. That's the good old Pythagorean identity there. And so when you have this thing rewritten, you can then split this as sine over cosine, sine of x over cosine x times 1 over cosine x dx. And the sine over cosine is tangent, and 1 over cosine is in fact secant. So let me just write it reverse-order, so it looks more familiar. This is secant of x times the tangent Of x dx. So now this question is rephrased as to what function has derivative secant tangent, and that is on our tables. That we should know from calc one, that is good old secant of x. Now remember, this notation means I want all the antiderivatives, so this is, don't forget the + C. So final answer here is secant x + C. Final answer in the original one, 2 sine of x + C. You will do more to get practice with these, but start with these and let me know if you have any questions. See you next time.
