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PROFESSOR: Today we're going
to continue our discussion

9
00:00:28,390 --> 00:00:33,040
of methods of integration.

10
00:00:33,040 --> 00:00:35,560
The method that I'm
going to describe today

11
00:00:35,560 --> 00:00:38,310
handles a whole
class of functions

12
00:00:38,310 --> 00:00:40,510
of the following form.

13
00:00:40,510 --> 00:00:53,460
You take P(x) / Q(x) and this
is known as a rational function.

14
00:00:53,460 --> 00:00:57,810
And all that means
is that it's a ratio

15
00:00:57,810 --> 00:01:09,680
of two polynomials, which are
these functions P(x) and Q(x).

16
00:01:13,300 --> 00:01:17,380
We'll handle all such
functions by a method which

17
00:01:17,380 --> 00:01:26,250
is known as partial fractions.

18
00:01:26,250 --> 00:01:36,190
And what this does is,
it splits P / Q into what

19
00:01:36,190 --> 00:01:45,260
you could call easier pieces.

20
00:01:45,260 --> 00:01:47,571
So that's going to be
some kind of algebra.

21
00:01:47,571 --> 00:01:49,070
And that's what
we're going to spend

22
00:01:49,070 --> 00:01:52,640
most of our time doing today.

23
00:01:52,640 --> 00:01:55,090
I'll start with an example.

24
00:01:55,090 --> 00:01:58,950
And all of my examples
will be illustrating

25
00:01:58,950 --> 00:02:01,830
more general methods.

26
00:02:01,830 --> 00:02:08,200
The example is to integrate the
function 1 1/(x-1) plus, say,

27
00:02:08,200 --> 00:02:15,240
3/(x+2), dx.

28
00:02:15,240 --> 00:02:17,400
That's easy to do.

29
00:02:17,400 --> 00:02:19,410
It's just, we already
know the answer.

30
00:02:19,410 --> 00:02:23,170
It's ln x - ln(x-1) + 3 ln(x+3).

31
00:02:26,780 --> 00:02:28,340
Plus a constant.

32
00:02:28,340 --> 00:02:37,050
So that's done.

33
00:02:37,050 --> 00:02:41,330
So now, here's the difficulty
that is going to arise.

34
00:02:41,330 --> 00:02:45,320
The difficulty is that I can
start with this function, which

35
00:02:45,320 --> 00:02:49,210
is perfectly manageable.

36
00:02:49,210 --> 00:02:52,830
And than I can add these
two functions together.

37
00:02:52,830 --> 00:02:55,380
The way I add fractions.

38
00:02:55,380 --> 00:03:00,750
So that's getting a
common denominator.

39
00:03:00,750 --> 00:03:06,140
And so that gives me
x+2 here, plus 3(x-1).

40
00:03:06,140 --> 00:03:11,090
And now if I combine
together all of these terms,

41
00:03:11,090 --> 00:03:16,960
then altogether I have
4x + 2 - 3, that's -1.

42
00:03:16,960 --> 00:03:23,260
And if I multiply out the
denominator that's x^2 plus--

43
00:03:23,260 --> 00:03:28,060
That 2 turned into a
3, that's interesting.

44
00:03:28,060 --> 00:03:31,240
I hope there aren't too many
more of those transformations.

45
00:03:31,240 --> 00:03:32,570
Is there another one here?

46
00:03:32,570 --> 00:03:36,830
STUDENT: [INAUDIBLE]

47
00:03:36,830 --> 00:03:39,150
PROFESSOR: Oh, it
happened earlier on.

48
00:03:39,150 --> 00:03:42,620
Wow that's an interesting
vibration there.

49
00:03:42,620 --> 00:03:45,510
OK.

50
00:03:45,510 --> 00:03:50,500
Thank you.

51
00:03:50,500 --> 00:03:56,770
So, I guess my 3's were
speaking to my 2's.

52
00:03:56,770 --> 00:03:58,580
Somewhere in my past.

53
00:03:58,580 --> 00:04:07,200
OK, anyway, I think
this is now correct.

54
00:04:07,200 --> 00:04:10,020
So the problem is the following.

55
00:04:10,020 --> 00:04:11,450
This is the problem with this.

56
00:04:11,450 --> 00:04:14,830
This integral was easy.

57
00:04:14,830 --> 00:04:18,500
I'm calling it easy, we
already know how to do it.

58
00:04:18,500 --> 00:04:19,450
Over here.

59
00:04:19,450 --> 00:04:28,040
But now over here,
it's disguised.

60
00:04:28,040 --> 00:04:31,947
It's the same function,
but it's no longer clear

61
00:04:31,947 --> 00:04:32,780
how to integrate it.

62
00:04:32,780 --> 00:04:34,580
If you're faced
with this one, you

63
00:04:34,580 --> 00:04:38,480
say, what am I supposed to do.

64
00:04:38,480 --> 00:04:41,790
And we have to get
around that difficulty.

65
00:04:41,790 --> 00:04:43,940
And so what we're
going to do is we're

66
00:04:43,940 --> 00:04:46,660
going to unwind this disguise.

67
00:04:46,660 --> 00:04:55,830
So we have the algebra
problem that we have.

68
00:04:55,830 --> 00:04:57,950
Oh, wow.

69
00:04:57,950 --> 00:04:59,510
There must be
something in the water.

70
00:04:59,510 --> 00:05:06,280
Impressive.

71
00:05:06,280 --> 00:05:06,920
Wow.

72
00:05:06,920 --> 00:05:08,350
OK, let's see.

73
00:05:08,350 --> 00:05:14,660
Is 2/3 = 3/2?

74
00:05:14,660 --> 00:05:17,960
Holy cow.

75
00:05:17,960 --> 00:05:18,760
Well that's good.

76
00:05:18,760 --> 00:05:21,340
Well, I'll keep you awake
today with several other

77
00:05:21,340 --> 00:05:23,940
transpositions here.

78
00:05:23,940 --> 00:05:30,760
So our algebra problem is to
detect the easy pieces which

79
00:05:30,760 --> 00:05:36,290
are inside.

80
00:05:36,290 --> 00:05:38,490
And the method that
we're going to use,

81
00:05:38,490 --> 00:05:40,330
the one that we'll
emphasize anyway,

82
00:05:40,330 --> 00:05:45,110
is one algebraic trick
which is a shortcut, which

83
00:05:45,110 --> 00:05:49,320
is called the cover-up method.

84
00:05:49,320 --> 00:05:51,900
But we're going to talk about
even more general things

85
00:05:51,900 --> 00:05:52,510
than that.

86
00:05:52,510 --> 00:05:54,300
But anyway, this is
where we're headed.

87
00:05:54,300 --> 00:05:57,910
Is something called
the cover-up method.

88
00:05:57,910 --> 00:05:59,490
Alright.

89
00:05:59,490 --> 00:06:02,640
So that's our intro.

90
00:06:02,640 --> 00:06:05,765
And I'll just have to
remember that 2 is not 3.

91
00:06:05,765 --> 00:06:08,130
I'll keep on repeating that.

92
00:06:08,130 --> 00:06:10,280
So now here I'm going
to describe to you

93
00:06:10,280 --> 00:06:13,880
how we unwind this disguise.

94
00:06:13,880 --> 00:06:17,660
The first step is, we
write down the function

95
00:06:17,660 --> 00:06:18,690
we want to integrate.

96
00:06:18,690 --> 00:06:21,890
Which was this.

97
00:06:21,890 --> 00:06:26,310
And now we have to undo the
first damage that we did.

98
00:06:26,310 --> 00:06:31,930
So the first step is to
factor the denominator.

99
00:06:31,930 --> 00:06:34,010
And that factors, we
happen to know the factors,

100
00:06:34,010 --> 00:06:36,150
so I'm not going
to carry this out.

101
00:06:36,150 --> 00:06:38,510
But this can be a
rather difficult step.

102
00:06:38,510 --> 00:06:41,110
But we're going to
assume that it's done.

103
00:06:41,110 --> 00:06:44,030
For the purposes of
illustration here.

104
00:06:44,030 --> 00:06:46,780
So I factor the denominator.

105
00:06:46,780 --> 00:06:50,630
And now, the second thing
that I'm going to do

106
00:06:50,630 --> 00:06:54,770
is what I'm going to
call the setup here.

107
00:06:54,770 --> 00:06:56,520
How I'm going to set things up.

108
00:06:56,520 --> 00:06:59,340
And I'll tell you what
these things are more

109
00:06:59,340 --> 00:07:01,010
systematically in a second.

110
00:07:01,010 --> 00:07:03,720
And the setup is that I
want to somehow detect

111
00:07:03,720 --> 00:07:05,240
what I did before.

112
00:07:05,240 --> 00:07:11,880
And I'm going to write
some unknowns here.

113
00:07:11,880 --> 00:07:15,650
What I expect is that this
will break up into two pieces.

114
00:07:15,650 --> 00:07:17,590
One with the
denominator x - 1, and

115
00:07:17,590 --> 00:07:23,470
the other with the
denominator x + 2.

116
00:07:23,470 --> 00:07:32,860
So now, my third step is going
to be to solve for A and B.

117
00:07:32,860 --> 00:07:34,500
And then I'm done, if I do that.

118
00:07:34,500 --> 00:07:41,370
That's the complete
unwinding of this disguise.

119
00:07:41,370 --> 00:07:43,940
And this is where the cover-up
method comes in handy.

120
00:07:43,940 --> 00:07:46,800
This is this method that
I'm about to describe.

121
00:07:46,800 --> 00:07:49,370
Now, you can do the
algebra in a clumsy way,

122
00:07:49,370 --> 00:07:50,970
or you can do it in a quick way.

123
00:07:50,970 --> 00:07:54,240
And we'd like to get efficient
about the algebra involved.

124
00:07:54,240 --> 00:07:58,590
And so let me show you what
the first step in the trick is.

125
00:07:58,590 --> 00:08:12,470
We're going to solve for A
by multiplying by (x - 1).

126
00:08:12,470 --> 00:08:16,045
Now, notice if you multiply
by (x - 1) in that equation 2,

127
00:08:16,045 --> 00:08:16,920
what you get is this.

128
00:08:16,920 --> 00:08:21,840
You got (4x - 2) divided
by-- The (x-1)'s cancel.

129
00:08:21,840 --> 00:08:23,920
You get this on
the left-hand side.

130
00:08:23,920 --> 00:08:26,480
And on the right-hand
side you get A.

131
00:08:26,480 --> 00:08:29,620
The (x-1)'s cancel again.

132
00:08:29,620 --> 00:08:31,420
And then we get this extra term.

133
00:08:31,420 --> 00:08:35,200
Which is B x+2 times x-1.

134
00:08:35,200 --> 00:08:39,200
Now, the trick here, and we're
going to get even better trick

135
00:08:39,200 --> 00:08:40,370
in just a second.

136
00:08:40,370 --> 00:08:42,870
The trick here is that
I didn't try to clear

137
00:08:42,870 --> 00:08:44,560
the denominators completely.

138
00:08:44,560 --> 00:08:46,820
I was very efficient
about the way I did it.

139
00:08:46,820 --> 00:08:51,390
It just cleared one factor.

140
00:08:51,390 --> 00:08:54,780
And the result here
is very useful.

141
00:08:54,780 --> 00:09:03,480
Namely, if I plug in now x =
1, this term drops out too.

142
00:09:03,480 --> 00:09:10,560
So what I'm going to do now
is I'm going to plug in x = 1.

143
00:09:10,560 --> 00:09:15,500
And what I get on the left-hand
side here is 4 - 1 and 1 + 2,

144
00:09:15,500 --> 00:09:19,740
and on the left-hand side
I get A. That's the end.

145
00:09:19,740 --> 00:09:27,150
This is my formula for A.
A happens to be equal to 1.

146
00:09:27,150 --> 00:09:29,100
And that's, of
course, what I expect.

147
00:09:29,100 --> 00:09:33,370
A had better be 1, because the
thing broke up into 1/(x-1) +

148
00:09:33,370 --> 00:09:33,870
3/(x+2).

149
00:09:37,100 --> 00:09:39,694
So this is the correct answer.

150
00:09:39,694 --> 00:09:41,610
There was a question out
here, which I missed.

151
00:09:41,610 --> 00:09:46,110
STUDENT: Aren't
polynomials defined

152
00:09:46,110 --> 00:09:56,210
as functions with whole powers,
or could they be square roots?

153
00:09:56,210 --> 00:09:57,650
PROFESSOR: Are
polynomials defined

154
00:09:57,650 --> 00:10:00,330
as functions with whole powers,
or can they be square roots?

155
00:10:00,330 --> 00:10:01,350
That's the question.

156
00:10:01,350 --> 00:10:04,140
The answer is, they
only have whole powers.

157
00:10:04,140 --> 00:10:06,950
So for instance here I only
have the power 1 and 0.

158
00:10:06,950 --> 00:10:10,410
Here I have the powers 2,
1 and 0 in the denominator.

159
00:10:10,410 --> 00:10:16,890
Square roots are no
good for this method.

160
00:10:16,890 --> 00:10:17,670
Another question.

161
00:10:17,670 --> 00:10:18,620
STUDENT: [INAUDIBLE]

162
00:10:18,620 --> 00:10:22,260
PROFESSOR: Why did I say x = 1?

163
00:10:22,260 --> 00:10:26,650
The reason why I said x = 1
was that it works really fast.

164
00:10:26,650 --> 00:10:30,400
You can't know this in advance,
that's part of the method.

165
00:10:30,400 --> 00:10:32,810
It just turns out to be
the best thing to do.

166
00:10:32,810 --> 00:10:36,130
The fastest way of getting
at the coefficient A. Now

167
00:10:36,130 --> 00:10:38,510
the curious thing, let me
just pause for a second

168
00:10:38,510 --> 00:10:39,320
before I do it.

169
00:10:39,320 --> 00:10:43,600
If I had plugged x = 1
into the original equation,

170
00:10:43,600 --> 00:10:45,070
I would have gotten nonsense.

171
00:10:45,070 --> 00:10:48,210
Because I would've gotten
0 in the denominator.

172
00:10:48,210 --> 00:10:50,940
And that seems like the
most horrible thing to do.

173
00:10:50,940 --> 00:10:54,280
The worst possible thing
to do, is to set x = 1.

174
00:10:54,280 --> 00:10:56,490
On the other hand,
what we did is a trick.

175
00:10:56,490 --> 00:10:58,870
We multiplied by x - 1.

176
00:10:58,870 --> 00:11:01,100
And that turned the
equation into this.

177
00:11:01,100 --> 00:11:05,810
So now, in disguise,
I multiplied by 0.

178
00:11:05,810 --> 00:11:07,890
But that turns out
to be legitimate.

179
00:11:07,890 --> 00:11:11,770
Because really this equation
is true for all x except 1.

180
00:11:11,770 --> 00:11:13,480
And then instead
of taking x = 1,

181
00:11:13,480 --> 00:11:15,580
I can really take x tends to 1.

182
00:11:15,580 --> 00:11:17,200
That's really what I need.

183
00:11:17,200 --> 00:11:18,530
The limit as x goes to one.

184
00:11:18,530 --> 00:11:20,610
The equation is
still valid then.

185
00:11:20,610 --> 00:11:22,760
So I'm using the worst
case, the case that

186
00:11:22,760 --> 00:11:24,360
looks like it's dividing by 0.

187
00:11:24,360 --> 00:11:26,760
And it's helping me because
it's cancelling out all

188
00:11:26,760 --> 00:11:31,730
the information in terms of B.
So the advantage here is this

189
00:11:31,730 --> 00:11:36,090
cancellation that
occurs in this part.

190
00:11:36,090 --> 00:11:37,440
So that's the method.

191
00:11:37,440 --> 00:11:39,930
We're going to shorten it
much, much more in a second.

192
00:11:39,930 --> 00:11:44,640
But let me carry it out for
the other coefficient as well.

193
00:11:44,640 --> 00:11:51,550
So the other coefficient
I'm going to solve for B,

194
00:11:51,550 --> 00:11:57,490
I'm going to multiply by x + 2.

195
00:11:57,490 --> 00:12:01,260
And when I do that,
I get (4x-1)/(x-1),

196
00:12:01,260 --> 00:12:05,020
that's the left-hand side,
the very top expression there.

197
00:12:05,020 --> 00:12:07,890
And then down below I
get A/(x-1) ( (x+2).

198
00:12:10,700 --> 00:12:14,380
And then again the
(x+2)'s cancel.

199
00:12:14,380 --> 00:12:15,960
So I get B sitting alone.

200
00:12:15,960 --> 00:12:18,150
And now I'm going to
do the same trick.

201
00:12:18,150 --> 00:12:21,540
I'm going to set x = -2.

202
00:12:21,540 --> 00:12:28,530
That's the value which is going
to knock out this A term here.

203
00:12:28,530 --> 00:12:30,770
So that cancels this
term completely.

204
00:12:30,770 --> 00:12:37,800
And what we get here all told
is - 8 - (-8 - 1)/(-2 - 1) = B.

205
00:12:37,800 --> 00:12:43,600
In other words, B = 3, which
was also what it was supposed

206
00:12:43,600 --> 00:12:44,100
to be.

207
00:12:44,100 --> 00:12:48,460
B was this number 3, right here.

208
00:12:48,460 --> 00:12:50,820
Which I'm now not
going to change to 2.

209
00:12:50,820 --> 00:12:52,496
Because I know that it's not 2.

210
00:12:52,496 --> 00:12:53,370
There was a question.

211
00:12:53,370 --> 00:12:59,540
STUDENT: [INAUDIBLE]

212
00:12:59,540 --> 00:13:01,420
PROFESSOR: All right.

213
00:13:01,420 --> 00:13:05,770
Now, this is the method
which is called cover-up.

214
00:13:05,770 --> 00:13:09,490
But it's really carried out
much, much faster than this.

215
00:13:09,490 --> 00:13:11,250
So I'm going to
review the method

216
00:13:11,250 --> 00:13:14,300
and I'm going to show you
what it is in general.

217
00:13:14,300 --> 00:13:23,030
So the first step is to factor
the denominator, Q. That's

218
00:13:23,030 --> 00:13:24,590
what I labeled 1 over there.

219
00:13:24,590 --> 00:13:29,350
That was the factorization
of the denominator up top.

220
00:13:29,350 --> 00:13:36,260
The second step is what I'm
going to call the setup.

221
00:13:36,260 --> 00:13:37,250
That's step 2.

222
00:13:37,250 --> 00:13:40,915
And that's where I knew what
I was aiming for in advance.

223
00:13:40,915 --> 00:13:42,540
And I'm going to have
to explain to you

224
00:13:42,540 --> 00:13:45,780
in every instance exactly
what this setup should be.

225
00:13:45,780 --> 00:13:49,070
That is, what the unknowns
should be and what target,

226
00:13:49,070 --> 00:13:53,080
simplified expression,
we're aiming for.

227
00:13:53,080 --> 00:13:54,950
So that's the setup.

228
00:13:54,950 --> 00:14:01,010
And then the third step is
what I'll now call cover-up.

229
00:14:01,010 --> 00:14:03,660
Which is just a very
fast way of doing

230
00:14:03,660 --> 00:14:05,720
what I did on this last
board, which is solving

231
00:14:05,720 --> 00:14:09,470
for the unknown coefficients.

232
00:14:09,470 --> 00:14:12,890
So now, let me perform
it for you again.

233
00:14:12,890 --> 00:14:14,170
Over here.

234
00:14:14,170 --> 00:14:18,990
So it's 4x - 1 divided by-- So
this is to eliminate writing

235
00:14:18,990 --> 00:14:19,650
here.

236
00:14:19,650 --> 00:14:20,190
Handwriting.

237
00:14:20,190 --> 00:14:24,130
It makes it much faster.

238
00:14:24,130 --> 00:14:28,490
So this part just
factoring the denominator,

239
00:14:28,490 --> 00:14:31,290
that was 1, that was step 1.

240
00:14:31,290 --> 00:14:34,050
And then step 2, again,
is the setup, which

241
00:14:34,050 --> 00:14:39,700
is setting it up like this.

242
00:14:39,700 --> 00:14:42,580
Alright, that's the setup.

243
00:14:42,580 --> 00:14:46,150
And now I claim that
without writing very much,

244
00:14:46,150 --> 00:14:49,350
I can figure out
what A and B are.

245
00:14:49,350 --> 00:14:51,350
Just by staring at this.

246
00:14:51,350 --> 00:14:53,670
So now what I'm going
to do is I'm just going

247
00:14:53,670 --> 00:14:55,720
to think what I did over there.

248
00:14:55,720 --> 00:14:57,570
And I'm just going
to do it directly.

249
00:14:57,570 --> 00:15:02,680
So let me show you what the
method consists of visually.

250
00:15:02,680 --> 00:15:08,950
I'm going to cover up, that
is, knock out this factor,

251
00:15:08,950 --> 00:15:13,560
and focus on this number here.

252
00:15:13,560 --> 00:15:15,610
And I'm going to plug
in the thing that

253
00:15:15,610 --> 00:15:17,580
makes this 0, which is x = 1.

254
00:15:17,580 --> 00:15:20,180
So I'm plugging in x = 1.

255
00:15:20,180 --> 00:15:21,670
To this left-hand side.

256
00:15:21,670 --> 00:15:28,195
And what I get is (4
- 1)/(1 + 2) = A. Now,

257
00:15:28,195 --> 00:15:29,820
that's the same thing
I did over there.

258
00:15:29,820 --> 00:15:33,610
I just did it by skipping the
intermediate algebra step,

259
00:15:33,610 --> 00:15:35,860
which is a lot of writing.

260
00:15:35,860 --> 00:15:38,580
So the cover-up method really
amounts to the following thing.

261
00:15:38,580 --> 00:15:40,385
I'm thinking of
multiplying this over here.

262
00:15:40,385 --> 00:15:43,240
It cancels this and it gets
rid of everything else.

263
00:15:43,240 --> 00:15:45,960
And it just leaves me with
A on the right-hand side.

264
00:15:45,960 --> 00:15:47,680
And I have to get rid
of it on this side.

265
00:15:47,680 --> 00:15:50,030
So in other words,
by eliminating this,

266
00:15:50,030 --> 00:15:53,090
I'm isolating a on
the right-hand side.

267
00:15:53,090 --> 00:15:56,170
So the cover-up is that I'm
covering this and getting

268
00:15:56,170 --> 00:15:57,670
A out of it.

269
00:15:57,670 --> 00:15:59,560
Similarly, I can
do the same thing

270
00:15:59,560 --> 00:16:04,720
with B. It's focused
on the value x = -2.

271
00:16:04,720 --> 00:16:07,010
And B is what I'm getting
on the right-hand side.

272
00:16:07,010 --> 00:16:10,400
And then I have
to cover up this.

273
00:16:10,400 --> 00:16:14,530
So if I cover up that, then
what's left over, with x = -2,

274
00:16:14,530 --> 00:16:16,720
is again (-8 - 1)/(-2 - 1).

275
00:16:22,580 --> 00:16:26,970
So this is the way the method
gets carried out in practice.

276
00:16:26,970 --> 00:16:32,070
Writing, essentially,
the least you can.

277
00:16:32,070 --> 00:16:36,580
Now, when you get to
several variables,

278
00:16:36,580 --> 00:16:42,170
it becomes just way more
convenient to do this.

279
00:16:42,170 --> 00:16:45,660
So now, let me just review
when cover-up works.

280
00:16:45,660 --> 00:16:55,980
So this cover-up
method works if Q(x)

281
00:16:55,980 --> 00:17:06,060
has distinct linear factors.

282
00:17:06,060 --> 00:17:11,780
And, so you need
two things here.

283
00:17:11,780 --> 00:17:13,840
It has to factor
completely, the denominator

284
00:17:13,840 --> 00:17:15,180
has to factor completely.

285
00:17:15,180 --> 00:17:20,250
And the degree of
the numerator has

286
00:17:20,250 --> 00:17:27,750
to be strictly less than the
degree of the denominator.

287
00:17:27,750 --> 00:17:30,080
I'm going to give
you an example here.

288
00:17:30,080 --> 00:17:36,010
So, for instance-- And this
tells you the general pattern

289
00:17:36,010 --> 00:17:38,070
of the setup also.

290
00:17:38,070 --> 00:17:44,090
Say you had x^2 +
3x + 8, let's say.

291
00:17:44,090 --> 00:17:46,020
Over (x - 1) (x - 2) (x + 5).

292
00:17:50,380 --> 00:17:52,920
So here I'm going to
tell you the setup.

293
00:17:52,920 --> 00:17:58,710
The setup is going to be -
A/(x-1) + B/(x-2) + C/(x+5).

294
00:18:02,820 --> 00:18:04,620
And it will always
break up into something.

295
00:18:04,620 --> 00:18:06,860
So however many
factors you have,

296
00:18:06,860 --> 00:18:09,770
you'll have to put in a
term for each of those.

297
00:18:09,770 --> 00:18:13,150
And then you can
find each number here

298
00:18:13,150 --> 00:18:26,060
by this cover-up method.

299
00:18:26,060 --> 00:18:29,450
Now we're done with that.

300
00:18:29,450 --> 00:18:33,280
And now we have to go on to
the algebraic complications.

301
00:18:33,280 --> 00:18:38,420
So would the first typical
algebraic complication be.

302
00:18:38,420 --> 00:18:50,760
It would be repeated
roots or repeated factors.

303
00:18:50,760 --> 00:18:56,850
Let me get one that doesn't come
out to be extremely ugly here.

304
00:18:56,850 --> 00:19:02,930
So this is what
we'll call Example 2.

305
00:19:02,930 --> 00:19:05,620
And this is going to
work when the degree,

306
00:19:05,620 --> 00:19:07,650
you always need that the
degree of the numerator

307
00:19:07,650 --> 00:19:12,060
is less than the degree
of the denominator.

308
00:19:12,060 --> 00:19:22,980
And Q has now repeated
linear factors.

309
00:19:22,980 --> 00:19:26,030
So let's see which example
I wanted to show you.

310
00:19:26,030 --> 00:19:28,830
So let's just give this here.

311
00:19:28,830 --> 00:19:34,080
I'll just repeat
the denominator.

312
00:19:34,080 --> 00:19:39,580
With an extra factor like this.

313
00:19:39,580 --> 00:19:41,440
Now, the main thing
you need to know,

314
00:19:41,440 --> 00:19:44,370
since I've already performed
the factorization for you.

315
00:19:44,370 --> 00:19:46,140
Already performed Step 1.

316
00:19:46,140 --> 00:19:49,177
This is Step 1 here.

317
00:19:49,177 --> 00:19:50,760
You have to factor
things all the way,

318
00:19:50,760 --> 00:19:53,910
and that's already
been done for you.

319
00:19:53,910 --> 00:19:56,840
And here's what this setup is.

320
00:19:56,840 --> 00:20:05,540
The setup is that it's of the
form A/(x-1) plus B/(x-1)^2 -

321
00:20:05,540 --> 00:20:11,550
we need another term for the
square here - plus C/(x+2).

322
00:20:15,590 --> 00:20:17,210
In general, if you
have more powers

323
00:20:17,210 --> 00:20:19,310
you just need to keep on
putting in those powers.

324
00:20:19,310 --> 00:20:24,140
You need one for
each of the powers.

325
00:20:24,140 --> 00:20:26,340
Why does it have to be squared?

326
00:20:26,340 --> 00:20:27,570
OK.

327
00:20:27,570 --> 00:20:28,680
Good question.

328
00:20:28,680 --> 00:20:31,660
So why in the world
am I doing this?

329
00:20:31,660 --> 00:20:37,070
Let me just give you one hint
as to why I'm doing this.

330
00:20:37,070 --> 00:20:41,600
It's very, very much like the
decimal expansion of a number

331
00:20:41,600 --> 00:20:44,820
or, say, the base 2
expansion of a number.

332
00:20:44,820 --> 00:21:00,410
So, for, example the number 7/16
is 0/2 + 1/2^2 + 1/2^3 plus,

333
00:21:00,410 --> 00:21:02,120
is that right?

334
00:21:02,120 --> 00:21:07,300
So it's 4/16 + 1/2^4.

335
00:21:07,300 --> 00:21:09,090
It's this sort of thing.

336
00:21:09,090 --> 00:21:11,880
And I'm getting this
power and this power.

337
00:21:11,880 --> 00:21:13,455
If I have higher
powers, I'm going

338
00:21:13,455 --> 00:21:14,705
to have to have more and more.

339
00:21:14,705 --> 00:21:17,310
So this is what happens
when I have a 2^4.

340
00:21:17,310 --> 00:21:20,550
I have to represent
things like this.

341
00:21:20,550 --> 00:21:24,430
That's what's coming out of
this piece with the repetitious

342
00:21:24,430 --> 00:21:24,930
here.

343
00:21:24,930 --> 00:21:26,390
Of the powers.

344
00:21:26,390 --> 00:21:31,070
This is just an analogy.

345
00:21:31,070 --> 00:21:31,906
Of what we're doing.

346
00:21:31,906 --> 00:21:33,280
Yeah, another
question over here.

347
00:21:33,280 --> 00:21:34,455
STUDENT: [INAUDIBLE]

348
00:21:34,455 --> 00:21:35,080
PROFESSOR: Yes.

349
00:21:35,080 --> 00:21:36,621
So this is an example,
but it's meant

350
00:21:36,621 --> 00:21:38,860
to represent the general
case and I will also

351
00:21:38,860 --> 00:21:41,210
give you a general picture.

352
00:21:41,210 --> 00:21:43,442
For sure, once you have
the second power here,

353
00:21:43,442 --> 00:21:45,400
you'll need both the
first and the second power

354
00:21:45,400 --> 00:21:46,504
mentioned over here.

355
00:21:46,504 --> 00:21:48,420
And since there's only
a first power over here

356
00:21:48,420 --> 00:21:52,060
I only have to mention a
first power over there.

357
00:21:52,060 --> 00:21:55,790
If this were a 3 here, there
would be one more term which

358
00:21:55,790 --> 00:22:00,630
would be the one for
(x-1)^3 in the denominator.

359
00:22:00,630 --> 00:22:03,940
That's what you just said.

360
00:22:03,940 --> 00:22:08,330
OK, now, what's different
about this setup

361
00:22:08,330 --> 00:22:11,670
is that the cover-up
method, although it works,

362
00:22:11,670 --> 00:22:13,120
it doesn't work so well.

363
00:22:13,120 --> 00:22:14,660
It doesn't work quite as well.

364
00:22:14,660 --> 00:22:29,790
The cover-up works for
the coefficients B and C,

365
00:22:29,790 --> 00:22:38,370
not A. We'll have a quick
method for the numbers B and C.

366
00:22:38,370 --> 00:22:40,530
To figure out what they are.

367
00:22:40,530 --> 00:22:42,630
But it will be a
little slower to get

368
00:22:42,630 --> 00:22:47,910
to A, which we will do last.

369
00:22:47,910 --> 00:22:56,780
Let me show you how it works.

370
00:22:56,780 --> 00:23:01,610
First of all, I'm going to do
the ordinary cover-up with C.

371
00:23:01,610 --> 00:23:05,610
So for C, I just want
to do the same old thing

372
00:23:05,610 --> 00:23:06,360
that I did before.

373
00:23:06,360 --> 00:23:09,370
I cover up this,
and that's going

374
00:23:09,370 --> 00:23:12,300
to get rid of all the junk
except for the C term.

375
00:23:12,300 --> 00:23:16,800
So I have to plug in x = -2.

376
00:23:16,800 --> 00:23:21,810
And I get x-- sorry, I get
(-2)^2 + 2 in the numerator.

377
00:23:21,810 --> 00:23:25,660
And I get (-2 - 1)^2
in the denominator.

378
00:23:25,660 --> 00:23:28,690
Remember I'm covering this up.

379
00:23:28,690 --> 00:23:30,790
So that's all there is
on the left-hand side.

380
00:23:30,790 --> 00:23:37,780
And on the right-hand
side all there is C.

381
00:23:37,780 --> 00:23:39,980
Everything else got killed
off, because it was x -

382
00:23:39,980 --> 00:23:40,580
2 times that.

383
00:23:40,580 --> 00:23:42,570
That's 0 times all
that other stuff.

384
00:23:42,570 --> 00:23:45,240
And the x - 2 over
here canceled.

385
00:23:45,240 --> 00:23:47,330
This is the shortcut
that I just described,

386
00:23:47,330 --> 00:23:50,920
and this is much faster than
doing all that arithmetic.

387
00:23:50,920 --> 00:23:52,620
And algebra.

388
00:23:52,620 --> 00:23:57,160
So all together this
is a 6/9, right?

389
00:23:57,160 --> 00:24:09,140
So it's C = 6/9, which is 2/3.

390
00:24:09,140 --> 00:24:12,640
Now, the other one
which is easy to do,

391
00:24:12,640 --> 00:24:14,730
I'm going to do by
the slow method first.

392
00:24:14,730 --> 00:24:17,350
But you omit a term.

393
00:24:17,350 --> 00:24:23,380
The idea is to cover up
the other bad factor.

394
00:24:23,380 --> 00:24:27,277
Cover-ups, I'll do it both
the fast way and the slow way.

395
00:24:27,277 --> 00:24:29,110
I'll do it the fast way
first, and then I'll

396
00:24:29,110 --> 00:24:30,440
show you the slow way.

397
00:24:30,440 --> 00:24:32,467
The fast way is
to cover this up.

398
00:24:32,467 --> 00:24:34,300
And then I have to cover
up everything else.

399
00:24:34,300 --> 00:24:35,740
That gets eliminated.

400
00:24:35,740 --> 00:24:40,950
And that includes everything
but B. So I get B on this side.

401
00:24:40,950 --> 00:24:42,970
And I get 1 on that side.

402
00:24:42,970 --> 00:24:45,790
So that's (1^2 + 2) / (1 + 2).

403
00:24:48,340 --> 00:24:56,250
So in other words, B = 1.

404
00:24:56,250 --> 00:24:58,710
That was pretty fast, so let
me show you what arithmetic

405
00:24:58,710 --> 00:25:00,060
was hiding behind that.

406
00:25:00,060 --> 00:25:01,690
What algebra was
hiding behind it.

407
00:25:01,690 --> 00:25:06,900
What I was really doing is this.

408
00:25:06,900 --> 00:25:16,390
And multiplying through
by (x-1)^2, so I got this.

409
00:25:16,390 --> 00:25:19,910
So this canceled here, so
this C just stands alone.

410
00:25:19,910 --> 00:25:21,790
And then I have here
C/(x+2) (x-1)^2.

411
00:25:26,290 --> 00:25:30,000
Notice again, I
cleared out that one--

412
00:25:30,000 --> 00:25:32,040
this term from the
denominator and sent it over

413
00:25:32,040 --> 00:25:33,600
to the other side.

414
00:25:33,600 --> 00:25:38,790
Now, what's happening
is that when I set x = 1

415
00:25:38,790 --> 00:25:43,050
here, this term is dying.

416
00:25:43,050 --> 00:25:44,730
This term is going
away, because there's

417
00:25:44,730 --> 00:25:47,840
more powers in the numerator
than in the denominator.

418
00:25:47,840 --> 00:25:50,070
This is still 0.

419
00:25:50,070 --> 00:25:52,330
And this one is gone also.

420
00:25:52,330 --> 00:25:58,220
So all that's left is B. Now,
I cannot pull that off with

421
00:25:58,220 --> 00:25:59,820
a single power of x - 1.

422
00:25:59,820 --> 00:26:02,250
I can't expose the A term.

423
00:26:02,250 --> 00:26:03,962
It's the B term
that I can expose.

424
00:26:03,962 --> 00:26:06,170
Because I can multiply
through by this thing squared.

425
00:26:06,170 --> 00:26:09,310
If I multiply through
by just x - 1,

426
00:26:09,310 --> 00:26:12,520
what'll happen here is I won't
have canceled this (x-1)^2.

427
00:26:12,520 --> 00:26:13,160
It's useless.

428
00:26:13,160 --> 00:26:15,020
I still have a 0
in the denominator.

429
00:26:15,020 --> 00:26:17,150
I'll have B / 0 when
I plug in x = 1.

430
00:26:17,150 --> 00:26:22,060
Which I can't use.

431
00:26:22,060 --> 00:26:32,950
Again, the cover-up method
is giving us B and C, not A.

432
00:26:32,950 --> 00:26:36,510
Now, for the last
term, for A, I'm

433
00:26:36,510 --> 00:26:40,380
going to just have to be
straightforward about it.

434
00:26:40,380 --> 00:26:51,260
And so I'll just suggest for A,
plug in your favorite number.

435
00:26:51,260 --> 00:26:59,510
So plug in my favorite number.

436
00:26:59,510 --> 00:27:01,430
Which is x = 0.

437
00:27:01,430 --> 00:27:04,617
And you won't be
able to plug in x = 0

438
00:27:04,617 --> 00:27:05,700
if you've already used it.

439
00:27:05,700 --> 00:27:09,770
Here the two numbers we've
already used are x = 1

440
00:27:09,770 --> 00:27:13,820
and x = -2.

441
00:27:13,820 --> 00:27:18,300
But we haven't used x =
0 yet, so that's good.

442
00:27:18,300 --> 00:27:21,400
I'm going to plug in now
x = 0 into the equation.

443
00:27:21,400 --> 00:27:22,450
What do I get?

444
00:27:22,450 --> 00:27:35,070
I get 0 (0^2 + 2) / ((-1)^2
* 2) is equal to, let's see.

445
00:27:35,070 --> 00:27:36,800
A is the thing
that I don't know.

446
00:27:36,800 --> 00:27:48,190
So it's A A/(-1) + B/(x-1)^2,
so B = 1, so that's 1 1/(-1)^2.

447
00:27:48,190 --> 00:27:51,760
And then C was 2/3.

448
00:27:51,760 --> 00:27:55,490
2/3 / (x+2).

449
00:27:55,490 --> 00:28:00,900
So that's 0 + 2.

450
00:28:00,900 --> 00:28:02,690
Don't give up at this point.

451
00:28:02,690 --> 00:28:03,900
This is a lot of algebra.

452
00:28:03,900 --> 00:28:06,290
You really have to plug
in all these numbers.

453
00:28:06,290 --> 00:28:08,400
You make one arithmetic
mistake and you're always

454
00:28:08,400 --> 00:28:09,649
going to get the wrong answer.

455
00:28:09,649 --> 00:28:16,790
This is very
arithmetically intensive.

456
00:28:16,790 --> 00:28:18,980
However, it does
simplify at this point.

457
00:28:18,980 --> 00:28:22,680
We have 2/2, that's 1.

458
00:28:22,680 --> 00:28:27,790
Is equal to -A + 1 + 1/3.

459
00:28:27,790 --> 00:28:29,050
So let's see.

460
00:28:29,050 --> 00:28:34,920
A on the other side,
this becomes A = 1/3.

461
00:28:34,920 --> 00:28:35,520
And that's it.

462
00:28:35,520 --> 00:28:36,600
This is the end.

463
00:28:36,600 --> 00:28:40,020
We've we've simplified
our function.

464
00:28:40,020 --> 00:28:48,280
And now it's easy to integrate.

465
00:28:48,280 --> 00:28:48,780
Question.

466
00:28:48,780 --> 00:28:49,488
Another question.

467
00:28:49,488 --> 00:29:01,927
STUDENT: [INAUDIBLE]

468
00:29:01,927 --> 00:29:04,510
PROFESSOR: So the question is,
if x = 0 has already been used,

469
00:29:04,510 --> 00:29:05,590
what do I do?

470
00:29:05,590 --> 00:29:10,024
And the answer is,
pick something else.

471
00:29:10,024 --> 00:29:11,440
And you said pick
a random number.

472
00:29:11,440 --> 00:29:13,400
And that's right, except
that if you really

473
00:29:13,400 --> 00:29:18,020
picked a random number
it would be 4.12567843,

474
00:29:18,020 --> 00:29:20,000
which would be difficult.
What you want to pick

475
00:29:20,000 --> 00:29:24,240
is the easiest possible
number you can think of.

476
00:29:24,240 --> 00:29:24,740
Yeah.

477
00:29:24,740 --> 00:29:32,710
STUDENT: [INAUDIBLE]

478
00:29:32,710 --> 00:29:37,050
PROFESSOR: If you had, as in
this sort of situation here,

479
00:29:37,050 --> 00:29:39,690
more powers.

480
00:29:39,690 --> 00:29:41,610
Wouldn't you have
more variables.

481
00:29:41,610 --> 00:29:42,900
Very good question.

482
00:29:42,900 --> 00:29:44,490
That's absolutely right.

483
00:29:44,490 --> 00:29:47,860
This was a 3 by 3
system in disguise,

484
00:29:47,860 --> 00:29:50,080
for these three
unknowns, A, B and C.

485
00:29:50,080 --> 00:29:52,770
What we started with in
the previous problem was

486
00:29:52,770 --> 00:29:53,950
two variables.

487
00:29:53,950 --> 00:29:56,190
It's over here, the
variables A and B.

488
00:29:56,190 --> 00:29:59,310
And as the degree of
the denominator goes up,

489
00:29:59,310 --> 00:30:02,900
the number of variables goes up.

490
00:30:02,900 --> 00:30:05,040
It gets more and more
and more complicated.

491
00:30:05,040 --> 00:30:06,820
More and more
arithmetically intensive.

492
00:30:06,820 --> 00:30:09,077
STUDENT: [INAUDIBLE]

493
00:30:09,077 --> 00:30:09,910
PROFESSOR: Well, so.

494
00:30:09,910 --> 00:30:11,535
The question is, how
would you solve it

495
00:30:11,535 --> 00:30:12,880
if you have two unknowns.

496
00:30:12,880 --> 00:30:16,610
That's exactly the point here.

497
00:30:16,610 --> 00:30:19,560
This is a system of simultaneous
equations for unknowns.

498
00:30:19,560 --> 00:30:23,180
And we have little tricks for
isolating single variables.

499
00:30:23,180 --> 00:30:25,880
Otherwise we're stuck with
solving the whole system.

500
00:30:25,880 --> 00:30:28,560
And you'd have to solve the
whole system by elimination,

501
00:30:28,560 --> 00:30:35,980
various other tricks.

502
00:30:35,980 --> 00:30:38,950
I'll say a little
more about that later.

503
00:30:38,950 --> 00:30:47,610
Now, I have to get one
step more complicated

504
00:30:47,610 --> 00:30:53,560
with my next example.

505
00:30:53,560 --> 00:31:02,860
My next example is going
to have a quadratic factor.

506
00:31:02,860 --> 00:31:05,760
So still I'm sticking to
the degree of the polynomial

507
00:31:05,760 --> 00:31:08,520
and the numerator is less than
the degree of the polynomial

508
00:31:08,520 --> 00:31:09,660
in the denominator.

509
00:31:09,660 --> 00:31:21,700
And I'm going to take the case
where Q has a quadratic factor.

510
00:31:21,700 --> 00:31:26,140
Let me just again
illustrate this by example.

511
00:31:26,140 --> 00:31:30,470
I have here (x - 1) (x^2 + 1).

512
00:31:30,470 --> 00:31:34,630
I'll make it about
as easy as they come.

513
00:31:34,630 --> 00:31:38,830
Now, the setup will be
slightly different here.

514
00:31:38,830 --> 00:31:40,510
Here's the setup.

515
00:31:40,510 --> 00:31:42,570
It's already factored.

516
00:31:42,570 --> 00:31:44,360
I've already done
as much as I can do.

517
00:31:44,360 --> 00:31:48,820
I can't factor this x^2 + 1 into
linear factors unless you know

518
00:31:48,820 --> 00:31:49,940
about complex numbers.

519
00:31:49,940 --> 00:31:51,600
If you know about
complex numbers

520
00:31:51,600 --> 00:31:53,170
this method becomes much easier.

521
00:31:53,170 --> 00:31:55,400
And it comes back to
the cover-up method.

522
00:31:55,400 --> 00:31:58,370
Which is the way that the
cover-up method was originally

523
00:31:58,370 --> 00:32:01,060
conceived by Heaviside.

524
00:32:01,060 --> 00:32:04,280
But you won't get
to that until 18.03.

525
00:32:04,280 --> 00:32:05,610
So we'll wait.

526
00:32:05,610 --> 00:32:08,360
This, by the way, is a method
which is used for integration.

527
00:32:08,360 --> 00:32:10,980
But it was invented
to do something

528
00:32:10,980 --> 00:32:13,565
with Laplace transforms
and inversion

529
00:32:13,565 --> 00:32:15,600
of certain kinds of
differential equations.

530
00:32:15,600 --> 00:32:17,020
By Heaviside.

531
00:32:17,020 --> 00:32:21,530
And so it came much
later than integration.

532
00:32:21,530 --> 00:32:26,360
But anyway, it's a
very convenient method.

533
00:32:26,360 --> 00:32:30,210
So here's the setup
with this one.

534
00:32:30,210 --> 00:32:34,440
Again, we want a term
for this (x - 1) factor.

535
00:32:34,440 --> 00:32:37,880
And now we're going to also have
a term with the denominator x^2

536
00:32:37,880 --> 00:32:39,120
+ 1.

537
00:32:39,120 --> 00:32:40,760
But this is the difference.

538
00:32:40,760 --> 00:32:44,590
It's now going to be a
first degree polynomial.

539
00:32:44,590 --> 00:32:53,040
One degree down from
the quadratic here.

540
00:32:53,040 --> 00:32:55,620
So this is what I keep
on calling the setup,

541
00:32:55,620 --> 00:32:57,250
this is number 2.

542
00:32:57,250 --> 00:32:59,870
You have to know that in
advance based on the pattern

543
00:32:59,870 --> 00:33:03,160
that you see on
the left-hand side.

544
00:33:03,160 --> 00:33:03,660
Yes.

545
00:33:03,660 --> 00:33:12,630
STUDENT: [INAUDIBLE]

546
00:33:12,630 --> 00:33:15,130
PROFESSOR: The question is, if
the degree of the numerator--

547
00:33:15,130 --> 00:33:18,290
So in this case,
if this were cubed,

548
00:33:18,290 --> 00:33:20,510
and this is matching with
the denominator, which

549
00:33:20,510 --> 00:33:23,590
is total of degree 3.

550
00:33:23,590 --> 00:33:26,910
The answer is that
this does not work.

551
00:33:26,910 --> 00:33:29,150
STUDENT: [INAUDIBLE]

552
00:33:29,150 --> 00:33:30,995
PROFESSOR: It
definitely doesn't work.

553
00:33:30,995 --> 00:33:33,370
And we're going to have to do
something totally different

554
00:33:33,370 --> 00:33:34,050
to handle it.

555
00:33:34,050 --> 00:33:37,180
Which turns out, fortunately,
to be much easier than this.

556
00:33:37,180 --> 00:33:41,850
But we'll deal with
that at the end.

557
00:33:41,850 --> 00:33:43,227
Keep this in mind.

558
00:33:43,227 --> 00:33:45,310
This is an easy way to
make a mistake if you start

559
00:33:45,310 --> 00:33:47,230
with a higher degree numerator.

560
00:33:47,230 --> 00:33:51,300
You'll never get
the right answer.

561
00:33:51,300 --> 00:33:54,870
So now, so I have my setup now.

562
00:33:54,870 --> 00:33:56,840
And now what can I do?

563
00:33:56,840 --> 00:33:58,960
Well, I claim that I
can still do cover-up

564
00:33:58,960 --> 00:34:05,790
for A. It's the same idea.

565
00:34:05,790 --> 00:34:07,680
I cover this guy up.

566
00:34:07,680 --> 00:34:10,460
And if I really multiply by it
it would knock everything out

567
00:34:10,460 --> 00:34:14,550
but A. So I cover this
up and I plug in x = 1.

568
00:34:14,550 --> 00:34:25,460
So I get here 1^2 / (1^2 +1)
= A. In other words, A = 1/2.

569
00:34:25,460 --> 00:34:28,560
Again cover-up is pretty
fast, as you can see.

570
00:34:28,560 --> 00:34:32,030
It's not too bad.

571
00:34:32,030 --> 00:34:41,040
Now, at this next stage,
I want to find B and C.

572
00:34:41,040 --> 00:34:45,990
And the best idea
is the slow way.

573
00:34:45,990 --> 00:34:48,120
Here, it's not too terrible.

574
00:34:48,120 --> 00:34:50,010
But it's just what
we're going to do.

575
00:34:50,010 --> 00:34:54,270
Which is to clear the
denominators completely.

576
00:34:54,270 --> 00:35:05,520
So for B and C, just
clear the denominator.

577
00:35:05,520 --> 00:35:07,645
That means multiply through
by that whole business.

578
00:35:07,645 --> 00:35:09,978
Now, when you do that on the
left-hand side you're going

579
00:35:09,978 --> 00:35:10,870
to get x^2.

580
00:35:10,870 --> 00:35:12,940
Because I got rid of
the whole denominator.

581
00:35:12,940 --> 00:35:16,850
On the right-hand side when
I bring this up, the x -

582
00:35:16,850 --> 00:35:18,600
1 will cancel with this.

583
00:35:18,600 --> 00:35:23,160
So the A term will
be A(x^2 + 1).

584
00:35:23,160 --> 00:35:29,230
And the Bx + C term will have
a remaining factor of x - 1.

585
00:35:29,230 --> 00:35:33,300
Because the x^2 + 1 will cancel.

586
00:35:33,300 --> 00:35:38,670
Again, the arithmetic
here is not too terrible.

587
00:35:38,670 --> 00:35:41,000
Now I'm going to
do the following.

588
00:35:41,000 --> 00:35:46,550
I'm going to look
at the x^2 term.

589
00:35:46,550 --> 00:35:49,580
On the left-hand side
and the right-hand side.

590
00:35:49,580 --> 00:35:51,970
And that will give me
one equation for B and C.

591
00:35:51,970 --> 00:35:54,490
And then I'm going to do the
same thing with another term.

592
00:35:54,490 --> 00:35:57,350
The x^2 term on the left-hand
side, the coefficient is 1.

593
00:35:57,350 --> 00:35:59,740
It's 1 times x^2.

594
00:35:59,740 --> 00:36:06,030
On the other side, it's A.
Remember I actually have A.

595
00:36:06,030 --> 00:36:08,870
So I'm going to put
it in, it's 1/2.

596
00:36:08,870 --> 00:36:11,250
So this is the A term.

597
00:36:11,250 --> 00:36:14,020
And so I get 1/2 x^2.

598
00:36:14,020 --> 00:36:18,640
And then the only other x^2 is
when this Bx multiplies this x.

599
00:36:18,640 --> 00:36:24,440
So Bx * x is Bx^2, so this is--
the other coefficient on x^2 is

600
00:36:24,440 --> 00:36:36,060
B. And that forces B to be 1/2.

601
00:36:36,060 --> 00:36:41,370
And last of all, I'm
going to do the x^0 term.

602
00:36:41,370 --> 00:36:44,530
Or, otherwise known
as the constant term.

603
00:36:44,530 --> 00:36:48,650
And on the left-hand side,
the constant term is 0.

604
00:36:48,650 --> 00:36:51,270
There is no constant term.

605
00:36:51,270 --> 00:36:57,310
On the right-hand side there's
a constant term, 1/2 * 1.

606
00:36:57,310 --> 00:36:58,520
That's 1/2 here.

607
00:36:58,520 --> 00:37:00,760
And then there's
another constant term,

608
00:37:00,760 --> 00:37:09,430
which is this constant
times this -1 Which is -C.

609
00:37:09,430 --> 00:37:17,952
And so the conclusion
here is that C = 1/2.

610
00:37:17,952 --> 00:37:18,660
Another question.

611
00:37:18,660 --> 00:37:19,160
Yeah.

612
00:37:19,160 --> 00:37:33,130
STUDENT: [INAUDIBLE]

613
00:37:33,130 --> 00:37:37,970
PROFESSOR: There's also
an x^0 hidden in here.

614
00:37:37,970 --> 00:37:40,890
Sorry, an x^1, that's what
you were asking about, sorry.

615
00:37:40,890 --> 00:37:42,400
There's also an x^1.

616
00:37:42,400 --> 00:37:46,050
The only reason why I didn't go
to the x^1 is that it turns out

617
00:37:46,050 --> 00:37:49,080
with these two I didn't need it.

618
00:37:49,080 --> 00:37:51,150
The other thing is
that by experience, I

619
00:37:51,150 --> 00:37:53,320
know that the extreme
ends of the multiplication

620
00:37:53,320 --> 00:37:54,690
are the easiest ends.

621
00:37:54,690 --> 00:37:57,150
And the middle terms
have tons of cross terms.

622
00:37:57,150 --> 00:37:59,750
And so I don't like the middle
term as much because it always

623
00:37:59,750 --> 00:38:01,130
involves more arithmetic.

624
00:38:01,130 --> 00:38:05,530
So I stick to the lowest and
the highest terms if I can.

625
00:38:05,530 --> 00:38:07,390
So that was really
a sneaky thing.

626
00:38:07,390 --> 00:38:10,460
I did that without
saying anything.

627
00:38:10,460 --> 00:38:10,960
Yes.

628
00:38:10,960 --> 00:38:14,095
STUDENT: [INAUDIBLE]

629
00:38:14,095 --> 00:38:15,470
PROFESSOR: Another
good question.

630
00:38:15,470 --> 00:38:17,120
Could I just set x equals 0?

631
00:38:17,120 --> 00:38:17,910
Absolutely.

632
00:38:17,910 --> 00:38:22,370
In fact, that's equivalent
to picking out the x^0 term.

633
00:38:22,370 --> 00:38:24,150
And you could plug in numbers.

634
00:38:24,150 --> 00:38:25,140
If you wanted.

635
00:38:25,140 --> 00:38:27,820
That's another way of doing
this besides doing that.

636
00:38:27,820 --> 00:38:30,930
So you can also plug in numbers.

637
00:38:30,930 --> 00:38:38,060
Can plug in numbers. x = 0.

638
00:38:38,060 --> 00:38:42,210
Actually, not x = 1, right?

639
00:38:42,210 --> 00:38:44,292
-1, 2, etc.

640
00:38:44,292 --> 00:38:46,000
Not 1 just because
we've already used it.

641
00:38:46,000 --> 00:38:48,270
We won't get interesting
information out.

642
00:38:48,270 --> 00:38:48,770
Yes.

643
00:38:48,770 --> 00:38:56,347
STUDENT: [INAUDIBLE]

644
00:38:56,347 --> 00:38:57,930
PROFESSOR: So the
question is, could I

645
00:38:57,930 --> 00:38:59,900
have done it this other way.

646
00:38:59,900 --> 00:39:02,880
With the polynomial,
with this other one.

647
00:39:02,880 --> 00:39:04,020
Yes, absolutely.

648
00:39:04,020 --> 00:39:05,436
So in other words
what I've taught

649
00:39:05,436 --> 00:39:09,466
you now is two choices which
are equally reasonable.

650
00:39:09,466 --> 00:39:10,840
The one that I
picked was the one

651
00:39:10,840 --> 00:39:14,105
that was the fastest
for this guy and the one

652
00:39:14,105 --> 00:39:15,980
that was fastest for
this one, but I could've

653
00:39:15,980 --> 00:39:19,040
done the other way around.

654
00:39:19,040 --> 00:39:22,490
There are a lot of ways of
solving simultaneous equations.

655
00:39:22,490 --> 00:39:23,777
Yeah, another question.

656
00:39:23,777 --> 00:39:24,610
STUDENT: [INAUDIBLE]

657
00:39:24,610 --> 00:39:26,318
PROFESSOR: The question
is the following.

658
00:39:26,318 --> 00:39:28,610
So now everybody can
understand the question.

659
00:39:28,610 --> 00:39:33,640
If this, instead of being
x^2 + 1, this were x^3 + 1.

660
00:39:33,640 --> 00:39:36,360
So that's an important
case to understand.

661
00:39:36,360 --> 00:39:39,420
That's a case in which
this denominator is not

662
00:39:39,420 --> 00:39:41,020
fully factored.

663
00:39:41,020 --> 00:39:46,830
So it's an x^3 + 1, you would
have to factor out an x + 1.

664
00:39:46,830 --> 00:39:49,630
So that would be a situation
like this, you have an x^3 + 1,

665
00:39:49,630 --> 00:39:57,740
but that's (x+1) (x^2 +x
+ 1), this kind of thing.

666
00:39:57,740 --> 00:40:01,830
If that's the right, there must
be a minus sign in here maybe.

667
00:40:01,830 --> 00:40:03,240
OK, something like this.

668
00:40:03,240 --> 00:40:07,274
Right?

669
00:40:07,274 --> 00:40:08,190
Isn't that what it is?

670
00:40:08,190 --> 00:40:11,561
STUDENT: [INAUDIBLE]

671
00:40:11,561 --> 00:40:12,810
PROFESSOR: I think it's right.

672
00:40:12,810 --> 00:40:15,930
But anyway, the point is
that you have to factor it.

673
00:40:15,930 --> 00:40:17,920
And then you have a
linear and a quadratic.

674
00:40:17,920 --> 00:40:20,880
So you're always going
to be faced eventually

675
00:40:20,880 --> 00:40:23,484
with linear factors
and quadratic factors.

676
00:40:23,484 --> 00:40:25,650
If you have a cubic, that
means you haven't factored

677
00:40:25,650 --> 00:40:28,620
sufficiently.

678
00:40:28,620 --> 00:40:31,947
So you're still back in Step 1.

679
00:40:31,947 --> 00:40:32,780
STUDENT: [INAUDIBLE]

680
00:40:32,780 --> 00:40:34,500
PROFESSOR: In the x^3 + 1 case?

681
00:40:34,500 --> 00:40:36,610
STUDENT: [INAUDIBLE]

682
00:40:36,610 --> 00:40:39,610
PROFESSOR: In the x^3 + 1
case, we are out of luck until

683
00:40:39,610 --> 00:40:41,700
we've completed
the factorization.

684
00:40:41,700 --> 00:40:43,470
Once we've completed
the factorization,

685
00:40:43,470 --> 00:40:46,370
we're going to have to
deal with these two factors

686
00:40:46,370 --> 00:40:48,920
as denominators.

687
00:40:48,920 --> 00:40:52,730
So it'll be this plus
something over x + 1

688
00:40:52,730 --> 00:40:58,160
plus a Bx + C type of
thing over this thing here.

689
00:40:58,160 --> 00:41:01,730
That's what's eventually
going to happen.

690
00:41:01,730 --> 00:41:03,940
But hold on to that idea.

691
00:41:03,940 --> 00:41:14,340
Let me carry out one
more example here.

692
00:41:14,340 --> 00:41:17,870
So I've figured out
what all the values are.

693
00:41:17,870 --> 00:41:21,420
But I think it's also
worth it to remember now

694
00:41:21,420 --> 00:41:29,140
that we also have to
carry out the integration.

695
00:41:29,140 --> 00:41:38,220
What I've just shown you is that
the integral of x^2 dx over (x

696
00:41:38,220 --> 00:41:43,180
- 1)(x^2 + 1) is equal to,
and I've split up into these

697
00:41:43,180 --> 00:41:43,970
pieces.

698
00:41:43,970 --> 00:41:45,070
So what are the pieces?

699
00:41:45,070 --> 00:42:01,560
The pieces are, 1/2, x -
1, plus 1/2 x / (x^2 + 1).

700
00:42:01,560 --> 00:42:02,990
This is the A term.

701
00:42:02,990 --> 00:42:04,400
This is the B term.

702
00:42:04,400 --> 00:42:12,460
And then there's the C term.

703
00:42:12,460 --> 00:42:16,340
So we'd better remember that we
know how to antidifferentiate

704
00:42:16,340 --> 00:42:18,300
these things.

705
00:42:18,300 --> 00:42:20,256
In other words, I want
to finish the problem.

706
00:42:20,256 --> 00:42:21,630
The others were
pretty easy, so I

707
00:42:21,630 --> 00:42:23,310
didn't bother to
finish my sentence,

708
00:42:23,310 --> 00:42:25,570
but here I want to
be careful and have

709
00:42:25,570 --> 00:42:28,850
you realize that there's
something a little more to do.

710
00:42:28,850 --> 00:42:31,340
First of all we have the,
the first one is no problem.

711
00:42:31,340 --> 00:42:35,240
That's this.

712
00:42:35,240 --> 00:42:39,910
The second one actually
is not too bad either.

713
00:42:39,910 --> 00:42:45,340
This is, by the advanced
guessing method,

714
00:42:45,340 --> 00:42:47,696
my favorite method,
something like the logarithm,

715
00:42:47,696 --> 00:42:50,070
because that's what's going
to appear in the denominator.

716
00:42:50,070 --> 00:42:51,770
And then, if you
differentiate this,

717
00:42:51,770 --> 00:42:53,620
you're going to
get 2x over this.

718
00:42:53,620 --> 00:42:55,030
But here we have 1/2.

719
00:42:55,030 --> 00:42:59,470
So altogether it's 1/4 of this.

720
00:42:59,470 --> 00:43:02,470
So I fixed the coefficient here.

721
00:43:02,470 --> 00:43:06,180
And then the last one,
you have to think back

722
00:43:06,180 --> 00:43:09,060
to some level of memorization
here and remember that this

723
00:43:09,060 --> 00:43:15,610
is 1/2 the arc tangent.

724
00:43:15,610 --> 00:43:20,870
STUDENT: [INAUDIBLE]

725
00:43:20,870 --> 00:43:22,200
PROFESSOR: Why did I go to 1/4?

726
00:43:22,200 --> 00:43:24,570
Because in disguise,
for this guy,

727
00:43:24,570 --> 00:43:35,140
I was thinking d/dx of ln(x^2
+ 1) is 2x / (x^2 + 1).

728
00:43:35,140 --> 00:43:39,880
Because it's the derivative
of this divided by itself.

729
00:43:39,880 --> 00:43:43,300
This is-- The derivative
of ln u is u' / u.

730
00:43:46,620 --> 00:43:50,030
(ln u)' = u' / u.

731
00:43:50,030 --> 00:43:54,350
That was what I applied.

732
00:43:54,350 --> 00:43:58,210
And what I had was 1/2, so I
need a total of 1/4 to cancel.

733
00:43:58,210 --> 00:44:06,640
So 2/4 is 1/2.

734
00:44:06,640 --> 00:44:09,990
Now I've got to get you
out of one more deep hole.

735
00:44:09,990 --> 00:44:12,900
And I'm going to save the
general pattern for next time.

736
00:44:12,900 --> 00:44:27,860
But I do want to
clarify one thing.

737
00:44:27,860 --> 00:44:30,040
So let's handle this thing.

738
00:44:30,040 --> 00:44:33,650
What if the degree
of P is bigger than

739
00:44:33,650 --> 00:44:41,570
or equal to the degree of Q.
That's the thing that I claimed

740
00:44:41,570 --> 00:44:42,810
was easier.

741
00:44:42,810 --> 00:44:45,880
And I'm going to describe
to you how it's done.

742
00:44:45,880 --> 00:44:48,940
Now, in analogy,
with numbers you

743
00:44:48,940 --> 00:44:55,070
might call this an
improper fraction.

744
00:44:55,070 --> 00:44:59,850
That's the thing that
should echo in your mind

745
00:44:59,850 --> 00:45:01,970
when you're thinking about this.

746
00:45:01,970 --> 00:45:04,600
And I'm just going to
do it by example here.

747
00:45:04,600 --> 00:45:06,800
Let's see., I
cooked up an example

748
00:45:06,800 --> 00:45:09,730
so that I don't
make an arithmetic

749
00:45:09,730 --> 00:45:11,220
mistake along the way.

750
00:45:11,220 --> 00:45:16,990
So there are two or three
steps that I need to explain.

751
00:45:16,990 --> 00:45:18,030
So here's an example.

752
00:45:18,030 --> 00:45:21,320
The denominator's degree 2,
the numerator is degree 3.

753
00:45:21,320 --> 00:45:24,610
It well exceeds, so
there's a problem here.

754
00:45:24,610 --> 00:45:27,880
Our method is not going to work.

755
00:45:27,880 --> 00:45:32,260
And the first step that
I want to carry out

756
00:45:32,260 --> 00:45:36,370
is to reverse Step 1.

757
00:45:36,370 --> 00:45:38,371
That is, I don't want
the factorization

758
00:45:38,371 --> 00:45:39,620
for what I'm going to do next.

759
00:45:39,620 --> 00:45:42,040
I want it multiplied out.

760
00:45:42,040 --> 00:45:48,670
That means I have to multiply
through, so I get x^2 + x - 2.

761
00:45:48,670 --> 00:45:52,980
I'm back to the
starting place here.

762
00:45:52,980 --> 00:45:56,510
And now, the next thing
that I'm going to do

763
00:45:56,510 --> 00:46:01,130
is, I'm going to
use long division.

764
00:46:01,130 --> 00:46:04,270
That's how you convert
an improper fraction

765
00:46:04,270 --> 00:46:07,432
to a proper fraction.

766
00:46:07,432 --> 00:46:08,890
This is something
you were supposed

767
00:46:08,890 --> 00:46:16,780
to learn in, I don't
know, Grade 4, I know.

768
00:46:16,780 --> 00:46:23,970
Grade 3, Grade 4,
Grade 5, Grade 6, etc.

769
00:46:23,970 --> 00:46:27,560
So here's how it works in
the case of polynomials.

770
00:46:27,560 --> 00:46:31,030
It's kind of amusing.

771
00:46:31,030 --> 00:46:37,760
So we're dividing this
polynomial into that one.

772
00:46:37,760 --> 00:46:41,950
And so the quotient to
first order here is x.

773
00:46:41,950 --> 00:46:44,300
That is, that's going
to match the top terms.

774
00:46:44,300 --> 00:46:48,500
So I get x^3 + x^2 - 2x.

775
00:46:48,500 --> 00:46:50,310
That's the product.

776
00:46:50,310 --> 00:46:51,890
And now I subtract.

777
00:46:51,890 --> 00:46:53,990
And it cancels.

778
00:46:53,990 --> 00:46:58,440
So we get here -x^2 + 2x.

779
00:46:58,440 --> 00:47:01,110
That's the difference.

780
00:47:01,110 --> 00:47:04,260
And now I need to divide
this next term in.

781
00:47:04,260 --> 00:47:08,190
And I need a -1.

782
00:47:08,190 --> 00:47:14,700
So -1 times this
is -x^2 - x + 2.

783
00:47:14,700 --> 00:47:16,260
And I subtract.

784
00:47:16,260 --> 00:47:17,800
And the x^2's cancel.

785
00:47:17,800 --> 00:47:24,210
And here I get +3x - 2.

786
00:47:24,210 --> 00:47:27,090
Now, this thing has a name.

787
00:47:27,090 --> 00:47:30,880
This is called the quotient.

788
00:47:30,880 --> 00:47:33,090
And this thing also has a name.

789
00:47:33,090 --> 00:47:39,070
This is called the remainder.

790
00:47:39,070 --> 00:47:43,530
And now I'll show you how
it works by sticking it

791
00:47:43,530 --> 00:47:44,560
into the answer here.

792
00:47:44,560 --> 00:47:47,250
The quotient is x - 1.

793
00:47:47,250 --> 00:47:52,010
And the remainder is,
let's get down there.

794
00:47:52,010 --> 00:47:53,810
(3x - 2) / (x^2 + x - 2).

795
00:47:58,820 --> 00:48:03,300
So the punchline here
is that this thing

796
00:48:03,300 --> 00:48:05,450
is easy to integrate.

797
00:48:05,450 --> 00:48:08,190
This is easy.

798
00:48:08,190 --> 00:48:13,350
And this one, you
can use, now you

799
00:48:13,350 --> 00:48:18,292
can use cover-up, The
method that we had before.

800
00:48:18,292 --> 00:48:19,750
Because the degree
of the numerator

801
00:48:19,750 --> 00:48:22,700
is now below the degree
of the denominator.

802
00:48:22,700 --> 00:48:25,920
It's now first degree and
this is second degree.

803
00:48:25,920 --> 00:48:27,870
What you can't do
is use cover-up

804
00:48:27,870 --> 00:48:29,950
to start out with here.

805
00:48:29,950 --> 00:48:32,630
That will give you
the wrong answer.

806
00:48:32,630 --> 00:48:36,952
So that's the end for today,
and see you next time.