WEBVTT

00:00:00.000 --> 00:00:08.950
JOEL LEWIS: Hi, welcome
back to recitation.

00:00:08.950 --> 00:00:11.260
In lecture, Professor
Miller did a bunch

00:00:11.260 --> 00:00:13.320
of examples of
integrals involving

00:00:13.320 --> 00:00:14.760
trigonometric functions.

00:00:14.760 --> 00:00:16.940
So I thought I would
give you a couple more,

00:00:16.940 --> 00:00:19.965
well I guess three more
examples, some of them

00:00:19.965 --> 00:00:22.090
are a little different
flavor than the ones he did,

00:00:22.090 --> 00:00:27.150
but some nice examples of some
integrals you can compute now.

00:00:27.150 --> 00:00:30.280
One thing you might
need for them is you're

00:00:30.280 --> 00:00:34.450
going to have to remember
some of your trig identities.

00:00:34.450 --> 00:00:37.350
Just like you had to remember
some of them last time.

00:00:37.350 --> 00:00:40.740
So in particular, one identity
that you might need today

00:00:40.740 --> 00:00:45.000
that you didn't need in lecture,
was the angle sum identity

00:00:45.000 --> 00:00:45.660
for cosine.

00:00:45.660 --> 00:00:47.310
So let me just
remind you what that

00:00:47.310 --> 00:00:51.400
is, it says the
cosine of a plus b

00:00:51.400 --> 00:01:01.791
is equal to cosine a cosine
b minus sine a sine b.

00:01:01.791 --> 00:01:03.290
So you're going to
need that formula

00:01:03.290 --> 00:01:07.130
to compute one of
these three integrals.

00:01:07.130 --> 00:01:08.660
So why don't you
pause the video,

00:01:08.660 --> 00:01:10.802
take some time to
work these out.

00:01:10.802 --> 00:01:13.010
Come back you can check your
answers against my work.

00:01:22.180 --> 00:01:24.390
Hopefully you had
some fun and some luck

00:01:24.390 --> 00:01:26.490
working on these integrals.

00:01:26.490 --> 00:01:29.130
Let's have a go at them.

00:01:29.130 --> 00:01:34.920
The first integral is sine
cubed x secant squared x.

00:01:34.920 --> 00:01:36.680
So I don't think
that Professor Miller

00:01:36.680 --> 00:01:40.640
did any integrals that
involved the secant function,

00:01:40.640 --> 00:01:43.710
but one thing to remember is
that the secant function is

00:01:43.710 --> 00:01:45.590
very closely related
to the cosine function.

00:01:45.590 --> 00:01:48.580
It's just cosine to the minus 1.

00:01:48.580 --> 00:01:55.850
Sorry, by that I mean 1 over
cosine, not the arccosine.

00:01:55.850 --> 00:01:58.530
So, in order to--

00:01:58.530 --> 00:02:01.960
so we can view this
first integral here

00:02:01.960 --> 00:02:07.040
as an integral of a power of
sine times a power of cosine,

00:02:07.040 --> 00:02:08.670
just like the ones
you had in lecture.

00:02:08.670 --> 00:02:12.770
So we can write this as
sine cubed of x times--

00:02:12.770 --> 00:02:19.360
I'm going to write cosine
x to the minus 2 dx.

00:02:19.360 --> 00:02:23.150
And here we see this is a nice
example where the sine occurs

00:02:23.150 --> 00:02:25.090
with an odd power.

00:02:25.090 --> 00:02:29.260
So when the sine occurs with an
odd power, or when one of them

00:02:29.260 --> 00:02:31.510
at least occurs
with an odd power,

00:02:31.510 --> 00:02:33.424
life is relatively simple.

00:02:33.424 --> 00:02:34.840
And what we do is
just what we did

00:02:34.840 --> 00:02:37.720
in lecture, which is we
break it up so that we just

00:02:37.720 --> 00:02:40.920
have one instance of the one
that occurs to an odd power

00:02:40.920 --> 00:02:44.430
and then a bunch of
powers of the other one.

00:02:44.430 --> 00:02:47.040
So in this case, we have
sine to an odd power.

00:02:47.040 --> 00:02:48.539
So we want to pull
out all the even,

00:02:48.539 --> 00:02:52.535
you know, all the extra
multiples, so in this case

00:02:52.535 --> 00:02:55.460
we have sine times sine squared.

00:02:55.460 --> 00:02:57.410
So we take all
those sine squareds

00:02:57.410 --> 00:03:00.980
and we convert them into
1 minus cosine squareds.

00:03:00.980 --> 00:03:06.866
This is going to be equal to the
integral of sine x times, well,

00:03:06.866 --> 00:03:12.060
times sine squared x,
which is 1 minus cosine

00:03:12.060 --> 00:03:21.360
squared x, times cosine x to
the minus second power dx.

00:03:21.360 --> 00:03:23.342
And now you take
all these cosines

00:03:23.342 --> 00:03:25.800
and you multiply them together
and you see what you've got.

00:03:25.800 --> 00:03:27.902
So in this case that's
equal to the integral-- OK,

00:03:27.902 --> 00:03:32.730
so 1 times cosine to the minus
2 is just cosine to the minus 2.

00:03:32.730 --> 00:03:39.870
So this first term is
cosine x to the minus 2

00:03:39.870 --> 00:03:42.980
sine x and then minus--
OK, we have cosine

00:03:42.980 --> 00:03:44.495
squared times cosine
to the minus 2,

00:03:44.495 --> 00:03:49.884
so that one's even simpler,
that's just minus sine x dx.

00:03:49.884 --> 00:03:51.300
So at this point,
if you like, you

00:03:51.300 --> 00:03:54.910
could do a substitution of
a trigonometric function.

00:03:54.910 --> 00:03:56.773
You could set u
equal to cosine x.

00:03:56.773 --> 00:03:58.006
All right?

00:03:58.006 --> 00:03:59.380
What's going to
happen if you set

00:03:59.380 --> 00:04:04.966
u equal to cosine x, well du
is going to be minus sine x dx.

00:04:08.040 --> 00:04:11.710
Each of these have exactly one
appearance of sine x in them,

00:04:11.710 --> 00:04:19.540
so this would become-- so
if we set u equal cosine x.

00:04:19.540 --> 00:04:25.050
So, as I said du
equals minus sine x dx.

00:04:25.050 --> 00:04:28.560
Because each of these
have a sine x dx in them,

00:04:28.560 --> 00:04:31.030
that's going to get
clumped into the du.

00:04:31.030 --> 00:04:34.520
And the cosines will just get
replaced with u everywhere.

00:04:34.520 --> 00:04:36.140
So I'm going to
take this over here

00:04:36.140 --> 00:04:38.190
so I have a little more space.

00:04:38.190 --> 00:04:42.960
So this becomes the integral
of u to the minus 2,

00:04:42.960 --> 00:04:49.080
I guess it's minus u to
the minus 2 plus 1 du.

00:04:49.080 --> 00:04:54.160
That minus sine x is
just exactly equal to du.

00:04:54.160 --> 00:04:54.660
OK.

00:04:57.640 --> 00:04:59.970
And now this is easy
to finish from here.

00:04:59.970 --> 00:05:03.150
This is equal to,
well OK, so minus 2,

00:05:03.150 --> 00:05:05.550
so we have to go
up one to minus 1,

00:05:05.550 --> 00:05:10.980
so that's just u to the
minus 1, plus 1 gives us

00:05:10.980 --> 00:05:15.350
a u, plus a constant,
and then of course we

00:05:15.350 --> 00:05:17.300
have to substitute back in.

00:05:17.300 --> 00:05:21.150
So this is u is cosine
x, so u to the minus 1

00:05:21.150 --> 00:05:28.490
is secant x plus cosine
x plus a constant.

00:05:28.490 --> 00:05:30.210
So that's the first one.

00:05:30.210 --> 00:05:32.680
For the second one,
sine x cosine of 2x.

00:05:37.790 --> 00:05:40.360
We actually have a slightly
more complicated situation here.

00:05:40.360 --> 00:05:43.270
Because the cosine
2x is no longer,

00:05:43.270 --> 00:05:46.200
it's not just a trigonometric
function of x alone,

00:05:46.200 --> 00:05:48.240
it's a trigonometric
function of 2x.

00:05:48.240 --> 00:05:52.290
So in order to do the sort of
the things we did in lecture,

00:05:52.290 --> 00:05:53.850
what you're going
to have to do is,

00:05:53.850 --> 00:05:56.520
you're going to have to expand
this out in terms of just sine

00:05:56.520 --> 00:05:59.710
x and cosine x, rather
than cosine of 2x.

00:05:59.710 --> 00:06:01.710
So in order to do that
you just need to remember

00:06:01.710 --> 00:06:02.940
your double angle formula.

00:06:02.940 --> 00:06:05.148
Now if you don't remember
your double angle formulas,

00:06:05.148 --> 00:06:08.300
one way you can remember them is
remember the angle sum formula

00:06:08.300 --> 00:06:12.360
and apply it with a
and b both equal to x.

00:06:12.360 --> 00:06:14.300
If you like.

00:06:14.300 --> 00:06:17.080
But, or you could just remember
your double angle formulas.

00:06:17.080 --> 00:06:19.890
So let's go over here.

00:06:19.890 --> 00:06:24.250
And so in this case, we have
the integral of sine x times,

00:06:24.250 --> 00:06:28.156
well cosine 2x-- so there
are actually, right,

00:06:28.156 --> 00:06:31.330
there are several different
ways you can write cosine 2x.

00:06:31.330 --> 00:06:34.560
So one way you can write it
is cosine squared x minus sine

00:06:34.560 --> 00:06:36.540
squared x, but we want
everything-- again

00:06:36.540 --> 00:06:38.642
we have an odd power of
sine, so it would be nice

00:06:38.642 --> 00:06:40.100
if everything else
could be written

00:06:40.100 --> 00:06:42.940
in terms of just cosine.

00:06:42.940 --> 00:06:47.045
So an equivalent form for
cosine 2x is to write,

00:06:47.045 --> 00:06:52.610
is that it's equal to 2
cosine squared x minus 1 dx.

00:06:52.610 --> 00:06:55.400
And here, this is very, very
similar to the last question.

00:06:55.400 --> 00:06:57.650
Again, you can make the
substitution u equals cosine,

00:06:57.650 --> 00:06:59.657
if you like.

00:06:59.657 --> 00:07:02.240
And so I'm not going to do out
the whole substitution for you,

00:07:02.240 --> 00:07:07.290
but this is going to work out
to 2 cosine squared x sine

00:07:07.290 --> 00:07:16.100
x, the integral of that is minus
2/3 cosine cubed x and then

00:07:16.100 --> 00:07:20.450
the integral of minus
sine x is cosine x

00:07:20.450 --> 00:07:23.790
plus a constant of integration.

00:07:23.790 --> 00:07:27.160
So that's the second one.

00:07:27.160 --> 00:07:30.760
The third one now is
of a similar flavor,

00:07:30.760 --> 00:07:35.954
except you have here, you
have sine 2x and cosine 3x.

00:07:35.954 --> 00:07:37.870
So again, when you want
to integrate something

00:07:37.870 --> 00:07:42.790
where you have-- so here we have
both trigonometric functions

00:07:42.790 --> 00:07:44.730
occur in multiples.

00:07:44.730 --> 00:07:46.570
Now, if it was
the same multiple,

00:07:46.570 --> 00:07:50.387
if it were 2x and 2x,
say, that would be fine.

00:07:50.387 --> 00:07:52.220
You know, you could
just make a substitution

00:07:52.220 --> 00:07:56.980
like u equals 2x or something
and then proceed as usual.

00:07:56.980 --> 00:07:58.840
But because they're
different multiples,

00:07:58.840 --> 00:08:01.050
we need, when we do
these trig integrals,

00:08:01.050 --> 00:08:04.860
we need to apply the methods
that we used in lecture.

00:08:04.860 --> 00:08:07.930
You want the arguments to agree.

00:08:07.930 --> 00:08:12.390
So you want it all to be
sine of something and cosine

00:08:12.390 --> 00:08:14.110
of the same thing.

00:08:14.110 --> 00:08:17.350
So here the thing
to do, I think,

00:08:17.350 --> 00:08:19.330
is to try and put
everything in terms

00:08:19.330 --> 00:08:21.720
of just sine x and cosine x.

00:08:21.720 --> 00:08:24.990
So sine 2x is one you should
remember, double angle formula.

00:08:24.990 --> 00:08:28.200
Cosine 3x, well in order to
figure out what cosine 3x is,

00:08:28.200 --> 00:08:30.660
you could just
know it, maybe you

00:08:30.660 --> 00:08:33.350
learned it once upon a time.

00:08:33.350 --> 00:08:35.710
The other thing you
can do for cosine 3x,

00:08:35.710 --> 00:08:39.420
is you can use this
angle sum formula.

00:08:39.420 --> 00:08:42.800
So let me just work that
out for you quickly.

00:08:45.310 --> 00:08:48.900
So cosine of 3x, well how do
you use the angle sum formula?

00:08:48.900 --> 00:08:54.070
You can write 3x as 2x plus x.

00:08:54.070 --> 00:08:59.050
So this is equal to
cosine of 2x plus x.

00:08:59.050 --> 00:09:02.790
And now this is a sum, so you
can use the cosine sum formula,

00:09:02.790 --> 00:09:12.660
so you get this is equal to
cosine 2x cosine x minus sine

00:09:12.660 --> 00:09:16.800
2x sine x.

00:09:16.800 --> 00:09:20.440
OK and now you can use
the double angle formulas

00:09:20.440 --> 00:09:22.320
here and here.

00:09:22.320 --> 00:09:25.378
so in the end, if you
do this all out, what

00:09:25.378 --> 00:09:28.140
you get is you're
going to get 4 cosine

00:09:28.140 --> 00:09:35.290
cubed x minus 3 cosine x.

00:09:35.290 --> 00:09:37.502
You might say, oh what
happened to the sine x's?

00:09:37.502 --> 00:09:38.960
We're going to get
a sine squared x

00:09:38.960 --> 00:09:41.360
and I want to write
everything in terms of cosine.

00:09:41.360 --> 00:09:44.690
So I've replaced the-- I've
done an extra step here

00:09:44.690 --> 00:09:46.420
that I'm not showing,
where I replaced

00:09:46.420 --> 00:09:49.540
the sine squared x with a
1 minus cosine squared x.

00:09:49.540 --> 00:09:51.830
So you make this
substitution, and you also

00:09:51.830 --> 00:09:54.320
make the substitution
for sine x-- sorry,

00:09:54.320 --> 00:09:56.538
for sine 2x that you have.

00:09:56.538 --> 00:10:00.970
OK, sine 2x is equal to-- you
know, double angle formula--

00:10:00.970 --> 00:10:04.380
2 sine x cosine x.

00:10:04.380 --> 00:10:07.350
And so when you multiply
sine 2x by cosine 3x

00:10:07.350 --> 00:10:11.660
you again have an expression
that's got powers of cosine

00:10:11.660 --> 00:10:14.769
with a single power of sine.

00:10:14.769 --> 00:10:17.310
So doing these simplifications,
you again get this nice form.

00:10:17.310 --> 00:10:21.780
One of them's odd,
the other one-- one

00:10:21.780 --> 00:10:23.770
of the trig functions,
sine or cosine

00:10:23.770 --> 00:10:26.212
appears to just the
power of 1, so, OK,

00:10:26.212 --> 00:10:28.520
so you can do this nice
simple substitution.

00:10:28.520 --> 00:10:31.380
You don't need to do any of
the double angle, or half angle

00:10:31.380 --> 00:10:32.846
complication.

00:10:32.846 --> 00:10:34.720
I'm not going to finish
this one out for you,

00:10:34.720 --> 00:10:39.200
but I did write down the answer
so you can check your answer.

00:10:39.200 --> 00:10:41.870
When you-- the third
integral that we had,

00:10:41.870 --> 00:10:51.500
which is integral sine 2x
cosine 3x dx, when you work it

00:10:51.500 --> 00:10:54.800
all out you should get,
let's see what I got, OK,

00:10:54.800 --> 00:11:06.430
you should get minus 8/5
cosine to the fifth x

00:11:06.430 --> 00:11:15.390
plus 2 cosine cubed x plus
a constant of integration.

00:11:15.390 --> 00:11:18.000
So if you work out
all the details,

00:11:18.000 --> 00:11:20.000
this is what you should get.

00:11:20.000 --> 00:11:21.382
I'll stop there.