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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.
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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.