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.