Welcome to calculus. I'm Professor Greist. We're about to begin lecture 29 on tables and software. In this course I've de-emphasized graphing calculators and other computational tools for solving calculus problems. However, when it comes to integration, there's only so much that I can teach you in this course. Many integral problems Or simply too difficult to be solved by hand and computers provide a wonderful tool. In this lesson, we'll give a brief introduction to some of these software based tools available and cover what they can and cannot do. Not all mathematics computations can or should be done with pencil and paper. I inherited from my father several old slide rules which are wonderful analogue devices for doing mathematics and I can remember my first calculator, simple though it was, soon to be replaced by more capable devices... But there are older technologies still, one of which is books. In particular, when doing a difficult integral, one might find it useful to go to an integral table. These typically appeared in the back of the big, thick calculus textbook. Let's see an example of how these tables might be used. Compute the integral of 3 d x over x minus 4 plus 4 over x. What one would typically do is go to the back of the book and scan up and down the table to look for some formula. That matched the form of what you're trying to solve. Now, this might work. It might not. If it doesn't, well, one would typically try to do some algebraic simplification, let's say in this case multiplying through numerator and denominator by x. In that case, then factoring the denominator gives something of the form x over x minus two quantity squared. Now that is something that does appear in our integral table. One has the integral of x over quantity ax plus b quantity squared And the formula follows from that. Now, I am sure that you could figure out how to do this with partial fractions, but let's use the table. In this case, what would one have? Well, b is negative 2 and a is equal to 1. And so, following the formula, one gets negative two over x minus two, plus log of x minus two, plus a constant. Now, wait, we have to multiply everything by that three that was out in front. And that's how one would use a table. Fortunately there are better methods available. Now with the advent of cheap and fast computation there are several software packages that are available for doing mathematics integrals in particular. Being something a bit more challenging then derivatives, we're going to focus on one of these called Wolfram Alpha. If you go to wolframalpha.com, then you'll see a screen come up that allows you to type in whatever your interested in exploring. You'll have to play around a little bit with some of the mathematics, notation involved. But it shouldn't be too unfamiliar. Let's do a central example in this case e to the x. And in this case, after a few moments of thinking, it will give us a bit of information. For example, it will give the graph of the function over various ranges. It will also tell us something about roots well, in this case, there's not much there the domain and the range. It will notably give Taylor expansions, and it will do so using Big O, so it's a good thing that we've learned that already. It will tell about derivatives and integrals, and other information as well including limits and various series expansions. Let's try a challenging integral and see what we get. We'll try to integrate sin cubed of x over two times cosine cubed of x over two. After thinking for a moment let's see what it comes up with. well it gives us an answer. 1 over 96 times quantity cosine 3x minus 9 cosine x. It even remembered the constant, that's wonderful. It will also give us graphs associated with this answer... Other forms of the integral very important in this case since the way that I would have done the problem might have led to a different looking answer. It will give series expansions again using bigger language. Now, in what I'm showing you here WolframAlpha file allows you to click the Show Steps button, unfortaunetly they changed that function alley and it's no longer available for free. You can however, pay for service which allows you to expand out all of the intermediate area steps and how to arrive. Let this answer, as you can imagine, is something that could be pretty useful. Let's consider a different example, lets see how hard we can make it and see what WolframAlpha will be able to do. [NOISE] Lets consider the integral of 1-X to the 7th. Third root minus one minus x cubed 7th root. And let's make this a definite integral. X going from zero to one. And let's see what happens in this case. well it's giving us an answer and that answer happens to be zero, but why? Well, WolframAlpha doesn't tell you why. But if you consider these two pieces, the seventh root of 1 minus x cubed and the cube root of 1 minus x to the 7th, with a little bit of thinking you'll see that these two pieces are inverses of one another. If you compose one end to the other then you'll get the identity back... That means that the graphs of these functions are symmetric about the line y equals x. And since we're going from zero to one, where it intersects the x axis, That means that the integral of the difference between these two must be 0. Because anything on the left is balanced out by the corresponding piece on the right. WolframAlpha does a great job but it doesn't explain the why. Let's say, that we wanted to solve that same integral. [NOISE]. But instead of making it a definite integral, we tried to type it in as an indefinite integral. Figuring, perhaps, we'll evaluate the limits and come up with the answer on our own. Well, in this case, the indefinite integral is now so simple. It's expressed in terms of hyper geometric functions of 2 variables. Well this is not a wrong answer but it's not exactly illuminating from where we're at right now. So like any tool you have to use it with caution and with intelligence. Let's consider different example, this one again a difficult Definite integral. The integral of sine to the n over quantity sine to the n plus cosine to the n. Notice that we didn't have to specify what our variable was in this case x, it intuits that we mean sine of x to the nth power etc. Let's evaluate this. As x goes from zero to pi over two, well after a little bit of thought and a little bit of more thought we get a properly interpreted question, but an answer that says no, not happening. Now, this is a free product, so we don't expect it to have super computer-like abilities, but let's try to work with what we have. I claim that one can show that the answer to this definite integral is pie over 4. This involves some tricky trigonometric formulae. I'm not going to show it to you. But let's say you suspect that this definite integral has a nice answer. What could you do? Well, let's try [SOUND] typing in something for a specific power, for a specific n. In this case, n equals 3. Then, WolframAlpha is able to handle that one very nicely. It gets not only the correct decimal answer, but the exact answer of this integral. Very good. Now, let's continue with a higher power still. In this case, n equals five. Well, at this point, WolframAlpha still gets the correct numerical Answer. But it no longer knows that that is really pi over four. And if we move to higher power still, well, things are going to break down. But whatever difficulties might arise, this and other computational tools. Are extremely useful. With a little bit of practice and some thinking, you can use this and other computational methods to solve problems. But more than that, you can use these tools as a means of exploring. Mathematics. In fact, you may discover new results or theorems. Computation is always pointed the way to new truths and new ideas and there is so much left to be done in mathematics. With these tools in hand You, too, might make a contribution. I encourage you to play with these tools. You've reached the end of chapter 3, but this is by no means the end of our reliance on integrals. Indeed, in chapter 4, we will turn to applications Of indefinite and definite integrals and c some classical and some modern ways in which these are used.