0:00

Welcome to Calculus. I'm Professor Ghrist.

We're about to begin lecture 28 on trigonometric integrals.

We're nearly done with chapter three. At this point we've amassed a large

collection of techniques for computing definite and indefinite integrals.

In this lesson what we're going to do is collect all of those methods together and

apply them to a particular class of integrals involving trigonometric

integrands. In this lesson we'll consider three

families of integrals involving trigonometric functions The first, powers

of sine and powers of cosine. The second, powers of tangent and powers

of secant. The third, multiples of sines and

cosines. Throughout, m and n are going to denote

positive integers. For our first class of integrals,

consider, as an integrand, sine to the m, cosine to the n.

Solving such a problem is going to break up into several different cases depending

on whether m and n are even or odd. This will give us a good chance to

practice our various integration techniques.

First of all if you're in the setting where m, the power of sine, is odd.

The the following substitution will be effective.

Expand sine squared as 1 minus cosine squared.

And then substitute n u equals cosign. This will reduce the integral to

something that is easily computed. Likewise if n, the power of cosine, is

odd you perform a similar simplification. Expand out cosine squared as 1 minus sine

squared, and then substitute u equals sine.

The one remaining case is when m and n are both even.

In this case, a substitution will not work.

A reduction formula is required. This tends to be a bit more involved and

difficult. This is the case that is going to cause

us a little bit of trouble. Let's consider a specific example of the

form sine to the m, cosine to the n. Where m is 5 and n is 4.

This puts us in a case where the power of sine is odd, therefore we can split off

an even number of powers of sine, and substitute in 1 minus cosine squared for

every sine squared that we have. Now, you see that there is one power of

sine left over. So that if we perform the substitution,

where u equals cosine theta. We have a copy of du minus sine theta, d

theta, sitting in the integrand. And we wind up getting minus the integral

of quantity 1 minus u squared, squared times u to the 4th, du.

This polynomial can be expanded out, and then easily integrated substituting back

in u equals cosine theta. Gives us a solution.

Negative cosine to the 5th over 5 plus 2 cosine to the 7th over 7 minus cosine to

the 9th over 9 plus a constant. Likewise if we had an odd power of

cosine. Then we could apply the same method where

we substitute in 1 minus sine squared for cosine squared and we have left over a

single power of cosine. Another u substitution will easily solve

this integral. Integrals involving powers of tangent and

secant follow a similar pattern depending on the parity of the powers.

If one is in the setting where the power of tangent is odd, then perform the

simplification expanding out every tangent squared as secant squared minus

1, then substitute in u equals secant. Likewise if n, the power of secant is

even then one can simplify every secant squared is 1 plus tangent squared.

The substitution u equals tangent will then work.

The one remaining case where a reduction formula is required, is when n the power

secant is odd, and m, the power of tangent is even.

Let's consider a example, in this case where the power of tangent is 5, and the

power of secant is 6. Then, because the power of secant is

even. One way to solve this integral is to

split off a secant squared, and replace the remaining even powers of secant with

quantity 1 plus tangent squared. This means that when we do the

substitution u equals tangent we have a copy of du sitting right there to be

absorbed. This yields a polynomial integral of the

form u to the 5th times quantity 1 plus u squared, squared.

By expanding that out integrating that polynomial and substituting back in

tangent for u, we easily obtain the answer, tangent to the 6th, over 6, plus

tangent to the 8th over 4, plus tangent to the 10th over 10, plus a constant.

However, there's another way to solve this integral as well.

Exploiting the fact that m, the power of tangent, is odd.

In this case, what we'll want to do is split off an even power of tangent,

substitute n secant squared minus 1 for tangent squared.

And then use the fact that there is a secant tangent left over.

To perform a substitution where u equals secant theta.

6:55

There remains, in each class that we've considered, one case where a simple

substitution does not work. This requires a reduction formula.

The first step is to simplify to sums of powers of cosine or secant.

For example if we look at the integral of sine to the 4th, cosine to the 4th.

This is in that one case where both powers are even.

We can replace sine squared with 1 minus cosine squared and obtain a sum of

integrals, each of which is an even power of cosine.

Then we can apply integration by parts twice to obtain the following general

reduction formula. The integral of cosine to the n is cosine

to the n minus 1, times sine, over n, plus n minus 1 over n, times the integral

of cosine to the n minus 2. This simplifies or reduces the level of

complexity of the integral involved. Likewise, for powers of secant, one can

express the integral of secant to the n as tangent secant to the n minus 2 over n

minus 1 plus n minus 2 over n minus 1 times the integral of secant to the n

minus 2. Now you do not have to memorize these

formulae, they're the type of thing that one looks up when you're stuck on a

difficult integral. Let's apply this reduction formula in the

specific case of the integral of cosine to the n.

Recalling the reduction formula for powers of cosine.

We're going to have to apply this iteratively many times until we get down

to a low enough power of cosine that we can proceed.

To make things a bit more concrete, let's do a definite integral, as theta goes

from negative pi over 2 to pi over 2. When we do so one of the things that's

nice is that the term cosine to the n minus 1 sine over n, in the reduction

formula, vanishes. When we perform evaluation.

So, for n greater than 1, we get that the definite integral from negative pi over 2

to pi over 2 of cosine to the n is n minus 1 over n times the same integral

with the power being n minus 2. This will allow us to come up with a

recursive solution. So let's look at all the different

powers. In the simplest case, where n equals 0,

well we can do that integral. The integral of d theta is theta

evaluated from negative pi over 2 to pi over 2 gives pi.

Likewise we can do n equals 1 explicitly, integrating cosine, getting sine,

evaluating at the limits gives us the value of 2.

Now, for higher powers we can use the reduction formula.

All we have to do is multiply by n minus 1 over n.

So, to get n equals 2, we multiply 2 minus 1 over 2.

That is 1 half times the integral in the case where n equals 0.

To get the value for n equals 3, we multiply the value for n equals 1 by 3

minus 1, over 3. We can continue for increasing values of

n, always looking back to n minus 2 and multiplying by n minus 1 over n.

11:09

Now let's look at this, what do you notice?

Well, first of all, it seems as though there's a real dependence on whether n is

even or odd. When n is even, there's a factor of pi

involved, and when n is odd, there's not. That's maybe not so surprising seeing

that even and odd powers have been different all throughout this lesson.

In general using a little bit of induction one can show that when n is

even the result of this integral is pi. Times 1 over 2 times 3 over 4 times 5

over 6 et cetera, all the way down to n minus 1 over n.

When n is odd, one obtains a similar looking result but starting with 2

instead of pi. And flipping things, the result is 2 3rds

times 4 5ths times 6 7ths all the way up to n minus 1 over n.

All of that times 2. By putting a 1 down in the denominator,

we can see a familiar pattern between these two definite Integrals.

Now, there's one last class of integrals that we're going to look at, and that is

something in the form the integral of sine of m theta times cosine of n theta.

The sine wave and the cosine wave have potentially different periods.

Now this integral requires an algebraic simplification.

The following formula is something that you do not have to know or have

memorized, but which is extremely useful. That is, sine of m theta times cosine of

n theta equals 1 half sine of m plus n theta, plus sine of m minus n theta.

At least in the case where m and n are not the same.

If we integrate both sides, with respect to theta, then we obtain the formula,

negative cosine m plus n, theta over 2 times quantity m plus n minus cosine of m

minus n theta over twice quantity m minus n.

And you can see here why m and n have to be different.

Likewise, one can do the same thing for sine of m theta times sine of n theta.

A similar reduction gives an integral that is computable.

Likewise, with a pair of cosines of different periods.

The result works out. Once again, these are not the kind of

formulae that you memorize, but they are useful to look up.

Now why would any of these integrals be useful to us?

One important reason is that sines and cosines, trigonometric functions pervade

the physical universe. If you go far enough in mathematics you

will learn about Fourier Analysis, which is a mathematics built on sines and

cosines. It's extremely useful in analyzing any

kind of wave. Whether acoustic or electromagnetic.

In fact one could argue that all of radar and signal processing is built on a basis

of integrals of sines and cosines. Now you know a bit about how to compute

them. I think you'll agree, some of those

integrals were kind of tricky. What do you do, when faced with harder

and harder integration problems? Do you have to employ ever more clever

tricks? No.

Mathematics is not about tricks. But rather, about principles.

However, when having to solve a difficult problem, we have to use every available

method. In our next lesson, we'll give a brief

introduction to those methods that are computer assisted.