Welcome to a unique little section about the Tables of Integrals, and this section goes against initially, you'd think of what we've been doing so far. But I'd like to think of it as enhancing it. But we're going to learn about the Tables of Integrals, and these are found all over the Internet or in the back of textbooks, and they just list tables and tables of known integrals. But here's the thing that you may or may not know by now is that only a small fraction of functions can be integrated. We think of like a pie chart with all possible functions on the world. These are a small piece. The ones that we're focusing on are the ones that we know off the top of our heads. These are the techniques that we've done, so we know these. Another small piece of the pie chart is, these are the ones that are integrable by methods that were not taught in this class. Integrable by advanced methods. Here's the weird part. The majority of functions are non integrable. We need to develop a new theory here that goes against the purist mathematicians in us. The ones that say like, we should all be pretty and it should be nice. We need to develop some new techniques. For the non-integral functions, we're going to see that you can numerically integrate or you can approximate, and this is of course, the more difficult functions, the ones that come up more often in real life that when you're working on complex problems. The known ones, these are the ones that you're seeing a lot in this class and the intergrable ones we will explore further. However, there are so many like just little techniques to things to do that it's nice to have a table of integrals and these are often difficult to figure out but the idea is you don't want someone figures it out if it is just tricky and not really a general strategy, we throw it up in the catalog and then it's for the world to use. Just to show you an example, let's do one here. I have the integral of 4 plus x squared all over x dx. Stare at this one for a second and imagine that you're trying to work this one out. I challenge you to work through it. It looks like when we've done, but we actually haven't done one like this. It's very complicated. Someone at one point figured this out. If you look at the Table of Integrals, there are, if you go into like the square root section, there's one that looks like a squared plus x squared over x. Or maybe they use u or something like that, whatever the variable is. In this case for us, a is two. Using the table of integrals, if you're allowed to use them, if that's encouraged, if that's the idea, I usually write TOY, Table of Integrals, just to communicate the fact that I'm using a Table of Integrals to find the antiderivative. Then you could find a solution to this thing. If you look at the Table of Integrals online or back of a book somewhere, then you can find that this is actually equal to the square root of a squared plus, I guess they use the variable squared. So a is two, somehow I should put 2 squared plus x squared minus 2, then times, which is going to move down here for a minute, natural log of the absolute value of 2 plus the square root of 2 squared, so four plus x squared all over x, and then plus c because it's a general antiderivative. That multiplication comes down here just so I don't run into each other. But this is the idea where you would look at this and say maybe this is a trig sub and you'd expect something with tangent, but something's going to go wrong but the fact that like tangent doesn't appear, and so this one is a natural log and it does this whole thing. It's a little bit messy, and you can use an expression of the [inaudible]. Table of Integrals have their time and they have their place. What's nice about it too is like as computers get better, obviously like the more complicated ones you'd go to it. But you have to put things in context here, that while computers can spit back to numbers, going through the math behind a developing techniques, helping out computers when they get stuck sometimes, are formulating national question, that's some of computers can't do. If you're looking for a decimal, sure computers are really good at that. Let's try just one more. Let's try one more, and what's nice about these, they look terrifying, so then they're coming. This is one where it's a Table of Integrals example. Sine squared x cosine of x, natural log of sine of x dx. This is where we're allowed to use the Table of Integrals, but if you search through the Table of Integrals, you're not going to find one that is exactly this way. You need a little bit of human intuition to help this one out. What we'll do instead is you look at this and hopefully by now you see a composition and you see its derivative as well. Let's make a substitution, u equals sine of x. Then we'll say that du is cosine of x dx. Well, let's make a substitution. In that case, I have sine squared, which becomes u squared times the natural log of u dx. This is one of the forms that we can find on the Table of Integrals. When you find this in this form, oddly enough, it's not of the form u squared, and u it's of a more general patterns. If you look at the Table of Integrals, here's another skill, you'll find this one. It's like u_n, natural log of u du is equal, and then it gives you the whole nasty solution, u_n plus 1 over n plus 1 squared brackets, parentheses n plus 1,ln of u minus 1, close the brackets plus c. So has this general form. In our case here, n is equal to 2. We use the Table of Integrals, and when we do that and we're submitting something, it's like citing your sources. You just let the reader know that you are using Table of Integrals, and now we can go ahead and plug things in. Our variable we start off with. This should be u, by the way, is u. We're going to say u_n plus 1, n is 2, that's becomes 3, over 3 squared, which is 9, the natural log of n plus 1. That's 3 ln of u minus 1, close it up plus c. Maybe they did a lot of integration by parts and figured this out, develop pattern. I don't know, but that's not what we're here to figure out. Our goal is to use the Table of Integrals to find the antiderivative of which we did, which is quite nice. Then remember you still have to answer in the variable you started with. Then we go back and put it in terms of x to get sine cubed over 9 brackets, 3 natural log of sine of x minus 1 plus c. As a check, you can take a derivative here. It's going to be nasty, but you will get back the thing you started with. Table of Integrals are there, there are a tool to be used and there's a time and a place to do it. Just some final thoughts on this section. Depending on the teacher, depending on the course, some people really like this section because it just allows you to integrate so many more and then some people don't like this section because it's like, "Well, why am I learning how to do this if there's tables for everything." But I want you to keep the big picture in mind that you're learning how to problem-solve in this class through integration as well as the applications of integration. You're learning about the tools as a carpenter, but then you're going to go off like build a house. I have seen the use to these tables usually in later courses. Right now I cut through the ideas like get really good at integrals. When you do more stuff with integrals, and hopefully by now you can see integrals are really useful. Particular for me when I teach in, taught when I was a student in Calc 3, you're doing all these different problems with physics and economics and all these different pieces to it. That the step of integration becomes like one step of many. The more challenging pieces the human have to do is create the problem, set up the integral, set up the bound, and then solve the problem. Now you see it's one step of many, and then to help you solve the problem, use a Table of Integrals as you need it. Ideas like a calculator or any website or wherever you want to use it as a tool, not a crutch. If it's a known integral, you supposed to know you don't have to run to the calculator for 2 plus 2 at this point, it won't be equivalent. You'll see some nasty integrals appear. No problem go to the Table of Integrals to solve it because you're doing all the other pieces, and then last but not least, you interpret the answer. Does the answer make sense? What are the units of the answer? Then how do I communicate this number back to whatever group I'm working with, whatever the client is, whatever the project is. There's so many pieces that go into problem-solving that when you're in Calc 2, sometimes you don't see that big picture where this is all headed. The Table of Integrals, they get lost as like an ends to mean, that's not really the case. You should know they exist. They're not on a test and that you can't use them on tests and stuff. You do need to show your work when you're submitting work, you show your work when you're doing any of these things. But for now, just know they exist and our hope is to use them as a tool, not a crutch. It'll be very clear in questions if we want you to use a table of integrals when you're doing these things, and of course, if you're ever not sure, then reach out to the instructor. That's enough for Tables of Integrals. See you next video.
