WEBVTT

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Welcome back to recitation.

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In this video, we're
going to be working

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on establishing the best
technique for finding

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an integral or finding
an antiderivative.

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We'll be doing this,
as you've seen probably

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in a lot of these
videos, in a row.

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And so in this
one in particular,

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we're going to
work on these two.

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So what I'd like us to
find, is for the letter A,

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I'd like us to find
an actual value

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if we take the integral from
minus 1 to 0 of this fraction.

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5 x squared minus 2x plus 3 over
the quantity x squared plus 1

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times x minus one.

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And then, the second
problem, we're

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just going to be finding
an antiderivative.

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So it's finding an
antiderivative of the function

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1 over x plus 1 times the square
root of negative x squared

00:00:49.490 --> 00:00:50.750
minus 2x.

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Now, this does have a domain
over which this function is

00:00:54.390 --> 00:00:59.065
well-defined, as long as
what's inside the square root

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is positive.

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And there are values of x
for which that's positive.

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I just didn't want us to have
to compute this one exactly.

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So we're just looking for
an antiderivative here.

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So the goal is, figure out
what strategy you want to use,

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work through that strategy,
and then I'll be back,

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and I'll show you which
strategy I picked,

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and the solution that I got.

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OK.

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Welcome back.

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Well, hopefully you were
able to make some headway

00:01:28.040 --> 00:01:29.090
in both of these.

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And so what we'll do
right away, is just

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we'll start with the first one.

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So on the first
one, we should be

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able to get an actual numerical
value at the conclusion

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of the problem.

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And if you look
at the first one,

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it's probably pretty obvious
you want to use partial fraction

00:01:42.730 --> 00:01:45.110
decomposition.

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You already have a
denominator that's factored,

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and so this is going to
be fairly easy to do.

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Now, what's one thing
you want to check

00:01:51.320 --> 00:01:52.778
with partial
fractions, is you want

00:01:52.778 --> 00:01:55.010
to make sure that the
degree of the numerator

00:01:55.010 --> 00:01:58.100
is smaller than the
degree of the denominator.

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Notice that the
numerator degree is 2

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and the denominator
degree is 3, because we

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have an x squared times x.

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So we don't have to
do any long division.

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We can just start the problem.

00:02:08.380 --> 00:02:12.450
So what I'm going to do, is I'm
going to actually decompose it

00:02:12.450 --> 00:02:14.250
without showing
you how I did it.

00:02:14.250 --> 00:02:16.604
And you've done that
practice enough,

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so I'm just going
to show you what

00:02:18.020 --> 00:02:20.870
I got with my decomposition,
and we'll go from there.

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So when I decompose,
for letter A,

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I actually get two integrals.

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And the first one I get is
the integral from minus 1

00:02:28.750 --> 00:02:35.800
to 0 of 2x over x
squared plus 1 dx.

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And the second one I get is
the integral, minus 1 to 0,

00:02:40.490 --> 00:02:45.276
of 3 over x minus 1 dx.

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Let me just double check
and make sure-- yes.

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That's what I got when I
did this problem earlier.

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So this is not magic.

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I actually did this already.

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That's how I got these.

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And now from here, we just have
to determine-- we just have

00:02:57.440 --> 00:02:58.940
to integrate both of these.

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Now, what would be the
strategy at this point?

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Well this one-- if this
had just been a 2 and no x,

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you'd be dealing with an
arctan type of problem.

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Because you'd be integrating
1 over x squared plus 1.

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But actually, because
we have this 2x here,

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this is really a
substitution problem.

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Right?

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The derivative of x
squared plus 1 is 2x.

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Right?

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So we see that really
what we're integrating

00:03:23.750 --> 00:03:26.230
is something like du over u.

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And so if you did this
substitution problem,

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you should get something
like natural log

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of x squared plus 1.

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Let me just double
check and make sure that

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gives me the derivative here.

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The derivative of natural
log of x squared plus 1

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is 1 over x squared
plus 1 times 2x.

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That gives me
exactly what's here.

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I don't need
absolute values here,

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because this is always positive.

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And I know I need to
evaluate it at minus 1 and 0.

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OK?

00:03:50.860 --> 00:03:52.830
So that takes care
of the left one.

00:03:52.830 --> 00:03:55.269
Now, the right one
is-- again, it's

00:03:55.269 --> 00:03:57.310
a pretty straightforward
one, because it's just 3

00:03:57.310 --> 00:03:58.490
over x minus 1.

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That's a natural log again.

00:04:00.160 --> 00:04:02.030
So this natural log
is even simpler.

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It's going to be 3 times the
natural log absolute x minus 1

00:04:07.710 --> 00:04:09.850
from minus 1 to 0.

00:04:09.850 --> 00:04:11.971
And so now we just have
to plug in everything.

00:04:11.971 --> 00:04:12.470
OK?

00:04:12.470 --> 00:04:17.190
So let's just do this one step
at a time, starting over here.

00:04:17.190 --> 00:04:19.390
So the natural log,
when I put in 0,

00:04:19.390 --> 00:04:20.970
I get the natural log of 1.

00:04:20.970 --> 00:04:23.160
That's 0.

00:04:23.160 --> 00:04:26.495
And I subtract what I get when
I put in negative 1 for x.

00:04:26.495 --> 00:04:28.670
And negative 1
squared gives me 1,

00:04:28.670 --> 00:04:32.530
so this is minus the
natural log of 2.

00:04:32.530 --> 00:04:35.000
And then I have plus 3
times whatever's over here.

00:04:35.000 --> 00:04:36.570
So now let's look at this.

00:04:36.570 --> 00:04:39.000
When I plug in 0, I
get natural log of 0

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minus 1, absolute value.

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That's natural log of 1.

00:04:41.280 --> 00:04:42.880
That's zero again.

00:04:42.880 --> 00:04:44.680
And then I get a minus.

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And then I put in negative 1.

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Negative 1 minus 1,
negative 2, absolute value,

00:04:49.350 --> 00:04:50.760
so it's natural log of 2.

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And so if I look it
at all the way across,

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I see I have a negative
natural log of 2

00:04:58.760 --> 00:05:01.410
and then I have 3
natural logs of 2.

00:05:01.410 --> 00:05:06.590
So the final answer is just
negative 4 natural log of 2.

00:05:06.590 --> 00:05:09.841
And that is where
we'll stop with (a).

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OK.

00:05:10.340 --> 00:05:13.940
So let me just remind you,
actually, before we go to (b).

00:05:13.940 --> 00:05:17.340
What we did in (a) was we did
partial fraction decomposition.

00:05:17.340 --> 00:05:20.820
And I gave you the numerators.

00:05:20.820 --> 00:05:24.260
And then on the first one, we
had to use maybe a substitution

00:05:24.260 --> 00:05:24.990
to figure it out.

00:05:24.990 --> 00:05:27.480
I didn't write explicitly
the substitution,

00:05:27.480 --> 00:05:29.810
but a substitution
gives us that integral,

00:05:29.810 --> 00:05:32.830
and this one is
directly a natural log.

00:05:32.830 --> 00:05:33.330
OK.

00:05:33.330 --> 00:05:36.830
Now let's look at (b).

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So (b)-- let me rewrite
the problem, because it's

00:05:43.140 --> 00:05:44.265
now a little far away.

00:05:47.250 --> 00:05:52.900
I think it's x plus 1 square
root of negative x squared

00:05:52.900 --> 00:05:55.420
minus 2x.

00:05:55.420 --> 00:05:56.550
OK.

00:05:56.550 --> 00:06:00.210
So (b), the reason--
I wanted to make sure

00:06:00.210 --> 00:06:03.430
we did a trig substitution
in a particular way,

00:06:03.430 --> 00:06:06.000
because I haven't
demonstrated those very much.

00:06:06.000 --> 00:06:08.400
So the denominator wound up
looking a little awkward,

00:06:08.400 --> 00:06:11.080
to force you to
do it in that way.

00:06:11.080 --> 00:06:13.750
But what we want to do, is
we want to actually complete

00:06:13.750 --> 00:06:14.960
the square on what's in here.

00:06:14.960 --> 00:06:17.100
And that might make you
a little bit nervous.

00:06:17.100 --> 00:06:20.330
But let me just do a little
sidebar work down here,

00:06:20.330 --> 00:06:22.630
and point out what we get.

00:06:22.630 --> 00:06:25.520
If we factor out
a negative here,

00:06:25.520 --> 00:06:29.340
we get an x squared plus 2x.

00:06:29.340 --> 00:06:29.840
OK?

00:06:29.840 --> 00:06:31.740
So we're going to complete
the square on the inside.

00:06:31.740 --> 00:06:33.406
Now this might make
some people nervous.

00:06:33.406 --> 00:06:35.800
They might say, you've a
negative under the square root.

00:06:35.800 --> 00:06:38.756
But I want to point out
that I have a negative here,

00:06:38.756 --> 00:06:40.130
but I could always
make x squared

00:06:40.130 --> 00:06:43.090
plus 2x a negative
number, and then

00:06:43.090 --> 00:06:46.910
I would have-- the negative
times a negative is a positive.

00:06:46.910 --> 00:06:48.950
For instance, I think
negative 1, right,

00:06:48.950 --> 00:06:52.890
if I put a negative 1 for
x, I get negative 2 plus 1

00:06:52.890 --> 00:06:54.220
is a negative value.

00:06:54.220 --> 00:06:55.907
So if I put a
negative 1 for x, I'm

00:06:55.907 --> 00:06:57.740
taking the square root
of a positive number.

00:06:57.740 --> 00:07:01.250
So there are values of x that
make this under the square root

00:07:01.250 --> 00:07:02.216
positive.

00:07:02.216 --> 00:07:02.715
OK?

00:07:02.715 --> 00:07:04.607
So you don't have
to worry about that.

00:07:04.607 --> 00:07:06.940
Now, if I want to complete
the square on what's in here,

00:07:06.940 --> 00:07:07.690
what do I do?

00:07:07.690 --> 00:07:10.000
I have x squared plus 2x.

00:07:10.000 --> 00:07:13.670
I obviously need to add a
1 to complete the square.

00:07:13.670 --> 00:07:14.370
Why is that?

00:07:14.370 --> 00:07:16.411
Because you take what's
here, you divide it by 2,

00:07:16.411 --> 00:07:17.270
and you square it.

00:07:17.270 --> 00:07:19.860
And so this actually
equals square root

00:07:19.860 --> 00:07:25.060
of negative x squared
plus 2x plus 1.

00:07:25.060 --> 00:07:27.900
And I want to subtract 1 so
that I haven't changed anything,

00:07:27.900 --> 00:07:31.531
but when I pull it out from the
negative, it's another plus 1.

00:07:31.531 --> 00:07:32.030
OK?

00:07:32.030 --> 00:07:33.710
Let's make sure we buy that.

00:07:33.710 --> 00:07:37.540
I've added 1 inside here, so
if I add 1 on the outside,

00:07:37.540 --> 00:07:40.490
this is actually a minus
1, and so this is a plus 1,

00:07:40.490 --> 00:07:42.420
so together they add up to 0.

00:07:42.420 --> 00:07:45.370
So I haven't changed
what's in the square root.

00:07:45.370 --> 00:07:48.100
So if I come back and put that
in right here, what do I get?

00:07:48.100 --> 00:07:55.590
I get the integral dx over
x plus 1 square root--

00:07:55.590 --> 00:07:56.824
let me move this over.

00:07:56.824 --> 00:07:58.490
I'm going to bring
this to the front-- 1

00:07:58.490 --> 00:08:03.310
minus x plus 1 squared.

00:08:03.310 --> 00:08:03.810
All right.

00:08:03.810 --> 00:08:06.670
So from here, we have to
do a trig substitution.

00:08:06.670 --> 00:08:08.420
Now, what trig
substitution we want to do,

00:08:08.420 --> 00:08:10.770
we can do sine or cosine.

00:08:10.770 --> 00:08:14.130
But I'm going to do cosine,
because I like secants

00:08:14.130 --> 00:08:18.000
better than cosecants, because
I have those memorized better.

00:08:18.000 --> 00:08:20.460
So that's why I'm
choosing cosine.

00:08:20.460 --> 00:08:24.580
You'll see why I chose
that way in a little bit.

00:08:24.580 --> 00:08:28.370
But it would be, you will get
the same answer if you do sine.

00:08:28.370 --> 00:08:28.870
OK.

00:08:28.870 --> 00:08:31.390
So I'm going to come
to the other side.

00:08:31.390 --> 00:08:32.120
Let's see.

00:08:32.120 --> 00:08:37.470
So I'm choosing cosine
theta equals x plus 1.

00:08:37.470 --> 00:08:39.650
That's the substitution
I'm making.

00:08:39.650 --> 00:08:41.620
And why am I making
that substitution?

00:08:41.620 --> 00:08:43.820
I'm making that
substitution because when

00:08:43.820 --> 00:08:47.140
I do 1 minus x plus 1
squared, that's actually

00:08:47.140 --> 00:08:50.540
1 minus cosine squared theta.

00:08:50.540 --> 00:08:53.840
So that's, this in here
is sine squared theta.

00:08:53.840 --> 00:08:56.644
And when I take the square
root, I just get a sine theta.

00:08:56.644 --> 00:08:58.310
So that should be
pretty familiar to you

00:08:58.310 --> 00:08:59.800
by now, this strategy.

00:08:59.800 --> 00:09:02.010
But the point I'm
making is that x plus 1

00:09:02.010 --> 00:09:04.920
will be a cosine theta,
and this whole square root

00:09:04.920 --> 00:09:06.300
is what becomes a sine theta.

00:09:06.300 --> 00:09:07.740
So you've seen
that a fair amount,

00:09:07.740 --> 00:09:09.770
but just to remind you that.

00:09:09.770 --> 00:09:13.000
And then the other thing we
need is to replace the dx.

00:09:13.000 --> 00:09:16.471
So the dx is going to
be, derivative of cosine

00:09:16.471 --> 00:09:18.720
is negative sine, so you're
going to get negative sine

00:09:18.720 --> 00:09:21.790
theta d theta.

00:09:21.790 --> 00:09:23.040
So now we know all the pieces.

00:09:23.040 --> 00:09:26.530
We said this was cosine, we
said the square root is sine,

00:09:26.530 --> 00:09:29.440
and the dx is
negative sine d theta.

00:09:29.440 --> 00:09:31.285
So let's rewrite that over here.

00:09:34.180 --> 00:09:36.740
So we have-- I'm going to
put the negative in front,

00:09:36.740 --> 00:09:38.640
so I don't have to
deal with it anymore.

00:09:38.640 --> 00:09:47.030
Negative sine theta over cosine
theta sine theta d theta.

00:09:47.030 --> 00:09:49.240
These divide out,
and I get negative 1

00:09:49.240 --> 00:09:53.330
over cosine theta, which is just
equal to negative secant theta.

00:09:53.330 --> 00:09:54.320
OK?

00:09:54.320 --> 00:09:55.727
So I have negative secant theta.

00:09:55.727 --> 00:09:57.060
Let me actually write that here.

00:09:57.060 --> 00:10:00.940
Negative integral of
secant theta d theta.

00:10:00.940 --> 00:10:02.200
And now what is that?

00:10:02.200 --> 00:10:05.820
Well, we know how
to integrate secant.

00:10:05.820 --> 00:10:09.180
So let me write that
in terms of theta.

00:10:09.180 --> 00:10:13.990
It's going to be negative
natural log absolute value

00:10:13.990 --> 00:10:17.670
secant theta plus tangent theta.

00:10:17.670 --> 00:10:19.800
And then we have
the plus c out here.

00:10:19.800 --> 00:10:21.990
What's the point of this?

00:10:21.990 --> 00:10:23.740
Well, we should maybe
have this memorized.

00:10:23.740 --> 00:10:25.823
If you have to look it up,
you have to look it up,

00:10:25.823 --> 00:10:27.080
but you saw this one in class.

00:10:27.080 --> 00:10:29.150
And the negative is
just dropping down here,

00:10:29.150 --> 00:10:31.040
so don't think I added
that negative in when

00:10:31.040 --> 00:10:32.360
I was taking antiderivative.

00:10:32.360 --> 00:10:33.571
It was already there.

00:10:33.571 --> 00:10:34.070
All right.

00:10:34.070 --> 00:10:34.930
So we're done.

00:10:34.930 --> 00:10:36.130
Oh, but we're not done.

00:10:36.130 --> 00:10:37.940
Why are we not done?

00:10:37.940 --> 00:10:40.392
We're not done, because we
started off with something

00:10:40.392 --> 00:10:42.850
in terms of x, and now we have
something in terms of theta,

00:10:42.850 --> 00:10:44.330
so we have to finish up.

00:10:44.330 --> 00:10:45.770
And how we do that,
is we go back,

00:10:45.770 --> 00:10:47.520
we look at the
substitution we made.

00:10:47.520 --> 00:10:50.280
If we make a triangle
based on that substitution,

00:10:50.280 --> 00:10:53.320
we figure out the values of
secant theta and tangent theta,

00:10:53.320 --> 00:10:55.834
and then we can plug
those in terms of x.

00:10:55.834 --> 00:10:57.375
So let's remind
ourselves-- I'm going

00:10:57.375 --> 00:10:59.060
to draw the triangle
in the middle here.

00:10:59.060 --> 00:11:01.630
Let's remind ourselves
of the relationship

00:11:01.630 --> 00:11:05.100
we had between theta and x.

00:11:05.100 --> 00:11:10.270
If this is theta, we said
cosine theta, right here,

00:11:10.270 --> 00:11:12.870
cosine theta was
equal to x plus 1.

00:11:12.870 --> 00:11:16.300
Cosine theta is adjacent
over hypotenuse.

00:11:16.300 --> 00:11:20.500
So we want to say, this is
x plus 1, and this is 1.

00:11:20.500 --> 00:11:22.880
And that implies by the
Pythagorean theorem,

00:11:22.880 --> 00:11:26.180
that this is square root of
1 minus quantity x plus 1

00:11:26.180 --> 00:11:27.856
squared.

00:11:27.856 --> 00:11:29.790
Let me move that over.

00:11:29.790 --> 00:11:31.790
Notice, then, this
also makes sense,

00:11:31.790 --> 00:11:33.300
why sine theta is what it is.

00:11:33.300 --> 00:11:36.000
Sine theta is this
value divided by 1.

00:11:36.000 --> 00:11:38.601
So that also helps
you understand that.

00:11:38.601 --> 00:11:39.100
All right.

00:11:39.100 --> 00:11:40.558
So now what do we
need to read off?

00:11:40.558 --> 00:11:43.390
We need to read off secant, and
we need to read off tangent.

00:11:43.390 --> 00:11:45.790
So secant is 1 over
cosine, so actually, we

00:11:45.790 --> 00:11:48.640
could have gotten that one
for free, from the cosine.

00:11:48.640 --> 00:11:52.730
So this 1 over cosine
is 1 over x plus 1.

00:11:52.730 --> 00:11:56.240
So this thing is equal
to negative natural log

00:11:56.240 --> 00:12:01.474
absolute value 1 over x plus
1 plus-- now, what's tangent?

00:12:01.474 --> 00:12:03.140
If I come back and
look at the triangle,

00:12:03.140 --> 00:12:06.910
tangent theta is
opposite over adjacent.

00:12:06.910 --> 00:12:07.460
Right?

00:12:07.460 --> 00:12:11.380
So I can actually just put it
all over x plus 1 if I wanted.

00:12:11.380 --> 00:12:13.310
But I already started
writing it separately,

00:12:13.310 --> 00:12:15.090
so I'll leave it like this.

00:12:15.090 --> 00:12:20.380
Square root of 1 minus x
plus 1 quantity squared.

00:12:20.380 --> 00:12:23.910
And then close that,
and then my plus c.

00:12:23.910 --> 00:12:26.520
So now I'm actually
finished with the problem.

00:12:26.520 --> 00:12:31.220
Because now I have an
antiderivative in terms of x.

00:12:31.220 --> 00:12:33.910
So let me just remind you where
this problem, where we started

00:12:33.910 --> 00:12:36.100
the problem, kind of
take us through quickly,

00:12:36.100 --> 00:12:37.340
and then we'll be done.

00:12:37.340 --> 00:12:39.750
So back to the
beginning, what we had,

00:12:39.750 --> 00:12:45.109
was we had an integral that
was a fractional problem,

00:12:45.109 --> 00:12:46.650
but we had an x plus
1 here, and then

00:12:46.650 --> 00:12:49.850
we had this really
messy-looking quadratic in here.

00:12:49.850 --> 00:12:52.480
To make it easy to deal with,
I factored out a negative sign,

00:12:52.480 --> 00:12:54.880
and then I saw I could
complete the square.

00:12:54.880 --> 00:12:57.270
Once you complete the
square, you actually

00:12:57.270 --> 00:12:59.450
get another x plus
1 in there, which

00:12:59.450 --> 00:13:03.080
helps us to see immediately, it
should be a trig substitution.

00:13:03.080 --> 00:13:05.030
So the substitution
that's natural to make,

00:13:05.030 --> 00:13:07.740
because you have a 1 minus
something involving an x,

00:13:07.740 --> 00:13:09.980
is going to be either
cosine or sine.

00:13:09.980 --> 00:13:11.724
I chose cosine.

00:13:11.724 --> 00:13:13.140
If you'd chosen
sine, you probably

00:13:13.140 --> 00:13:15.270
would have gotten a
cosecant up there,

00:13:15.270 --> 00:13:18.804
instead of a secant, when you
were taking an antiderivative

00:13:18.804 --> 00:13:19.470
at the very end.

00:13:19.470 --> 00:13:22.280
So you would have gotten
the same answer because

00:13:22.280 --> 00:13:25.440
of the substitutions in the end.

00:13:25.440 --> 00:13:28.370
But so I chose cosine
theta is equal to x plus 1.

00:13:28.370 --> 00:13:29.820
You do that, you
can replace this

00:13:29.820 --> 00:13:33.630
with cosine, this with a sine,
this becomes a negative sine,

00:13:33.630 --> 00:13:35.450
and then you start simplifying.

00:13:35.450 --> 00:13:37.790
So once we came over
here and simplified,

00:13:37.790 --> 00:13:39.720
we got it into
something we recognize.

00:13:39.720 --> 00:13:41.140
We got it into secant.

00:13:41.140 --> 00:13:44.780
We know the
antiderivative for secant,

00:13:44.780 --> 00:13:46.110
in terms of secant and tangent.

00:13:46.110 --> 00:13:47.730
We know it's exactly this.

00:13:47.730 --> 00:13:50.730
And then we went back to
the relationship we had.

00:13:50.730 --> 00:13:53.470
We made ourselves
a triangle in terms

00:13:53.470 --> 00:13:55.130
of the theta and the x-values.

00:13:55.130 --> 00:13:57.130
And then we were
able to substitute in

00:13:57.130 --> 00:13:59.300
for secant and tangent.

00:13:59.300 --> 00:14:00.040
All right.

00:14:00.040 --> 00:14:02.140
So hopefully that was
successful for you.

00:14:02.140 --> 00:14:04.145
And that's where I'll stop.