1:09

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.

4:56

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.