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But there are older technologies still, one of which is books.

Â In particular, when doing a difficult integral, one might find it useful to go

Â to an integral table. These typically appeared in the back of

Â the big, thick calculus textbook. Let's see an example of how these tables

Â might be used. Compute the integral of 3 d x over x

Â minus 4 plus 4 over x. What one would typically do is go to the

Â back of the book and scan up and down the table to look for some formula.

Â That matched the form of what you're trying to solve.

Â Now, this might work. It might not.

Â If it doesn't, well, one would typically try to do some algebraic simplification,

Â let's say in this case multiplying through numerator and denominator by x.

Â In that case, then factoring the denominator gives something of the form x

Â over x minus two quantity squared. Now that is something that does appear in

Â our integral table. One has the integral of x over quantity

Â ax plus b quantity squared And the formula follows from that.

Â Now, I am sure that you could figure out how to do this with partial fractions,

Â but let's use the table. In this case, what would one have?

Â Well, b is negative 2 and a is equal to 1.

Â And so, following the formula, one gets negative two over x minus two, plus log

Â of x minus two, plus a constant. Now, wait, we have to multiply everything

Â by that three that was out in front. And that's how one would use a table.

Â Fortunately there are better methods available.

Â Now with the advent of cheap and fast computation there are several software

Â packages that are available for doing mathematics integrals in particular.

Â Being something a bit more challenging then derivatives, we're going to focus on

Â one of these called Wolfram Alpha. If you go to wolframalpha.com, then

Â you'll see a screen come up that allows you to type in whatever your interested

Â in exploring. You'll have to play around a little bit

Â with some of the mathematics, notation involved.

Â But it shouldn't be too unfamiliar. Let's do a central example in this case e

Â to the x. And in this case, after a few moments of

Â thinking, it will give us a bit of information.

Â For example, it will give the graph of the function over various ranges.

Â It will also tell us something about roots well, in this case, there's not

Â much there the domain and the range. It will notably give Taylor expansions,

Â and it will do so using Big O, so it's a good thing that we've learned that

Â already. It will tell about derivatives and

Â integrals, and other information as well including limits and various series

Â expansions. Let's try a challenging integral and see

Â what we get. We'll try to integrate sin cubed of x

Â over two times cosine cubed of x over two.

Â 4:56

After thinking for a moment let's see what it comes up with.

Â well it gives us an answer. 1 over 96 times quantity cosine 3x minus

Â 9 cosine x. It even remembered the constant, that's

Â wonderful. It will also give us graphs associated

Â with this answer... Other forms of the integral very

Â important in this case since the way that I would have done the problem might have

Â led to a different looking answer. It will give series expansions again

Â using bigger language. Now, in what I'm showing you here

Â WolframAlpha file allows you to click the Show Steps button, unfortaunetly they

Â changed that function alley and it's no longer available for free.

Â You can however, pay for service which allows you to expand out all of the

Â intermediate area steps and how to arrive.

Â Let this answer, as you can imagine, is something that could be pretty useful.

Â Let's consider a different example, lets see how hard we can make it and see what

Â WolframAlpha will be able to do. [NOISE] Lets consider the integral of 1-X

Â to the 7th. Third root minus one minus x cubed 7th

Â root. And let's make this a definite integral.

Â X going from zero to one. And let's see what happens in this case.

Â well it's giving us an answer and that answer happens to be zero, but why?

Â Well, WolframAlpha doesn't tell you why. But if you consider these two pieces, the

Â seventh root of 1 minus x cubed and the cube root of 1 minus x to the 7th, with a

Â little bit of thinking you'll see that these two pieces are inverses of one

Â another. If you compose one end to the other then

Â you'll get the identity back... That means that the graphs of these

Â functions are symmetric about the line y equals x.

Â And since we're going from zero to one, where it intersects the x axis, That

Â means that the integral of the difference between these two must be 0.

Â Because anything on the left is balanced out by the corresponding piece on the

Â right. WolframAlpha does a great job but it

Â doesn't explain the why. Let's say, that we wanted to solve that

Â same integral. [NOISE].

Â But instead of making it a definite integral, we tried to type it in as an

Â indefinite integral. Figuring, perhaps, we'll evaluate the

Â limits and come up with the answer on our own.

Â Well, in this case, the indefinite integral is now so simple.

Â It's expressed in terms of hyper geometric functions of 2 variables.

Â Well this is not a wrong answer but it's not exactly illuminating from where we're

Â at right now. So like any tool you have to use it with

Â caution and with intelligence. Let's consider different example, this

Â one again a difficult Definite integral. The integral of sine to the n over

Â quantity sine to the n plus cosine to the n.

Â Notice that we didn't have to specify what our variable was in this case x, it

Â intuits that we mean sine of x to the nth power etc.

Â Let's evaluate this. As x goes from zero to pi over two, well

Â after a little bit of thought and a little bit of more thought we get a

Â properly interpreted question, but an answer that says no, not happening.

Â Now, this is a free product, so we don't expect it to have super computer-like

Â abilities, but let's try to work with what we have.

Â I claim that one can show that the answer to this definite integral is pie over 4.

Â This involves some tricky trigonometric formulae.

Â I'm not going to show it to you. But let's say you suspect that this

Â definite integral has a nice answer. What could you do?

Â Well, let's try [SOUND] typing in something for a specific power, for a

Â specific n. In this case, n equals 3.

Â Then, WolframAlpha is able to handle that one very nicely.

Â It gets not only the correct decimal answer, but the exact answer of this

Â integral. Very good.

Â Now, let's continue with a higher power still.

Â In this case, n equals five. Well, at this point, WolframAlpha still

Â gets the correct numerical Answer. But it no longer knows that that is

Â really pi over four. And if we move to higher power still,

Â well, things are going to break down. But whatever difficulties might arise,

Â this and other computational tools. Are extremely useful.

Â With a little bit of practice and some thinking, you can use this and other

Â computational methods to solve problems. But more than that, you can use these

Â tools as a means of exploring. Mathematics.

Â In fact, you may discover new results or theorems.

Â Computation is always pointed the way to new truths and new ideas and there is so

Â much left to be done in mathematics. With these tools in hand You, too, might

Â make a contribution. I encourage you to play with these tools.

Â