1
00:00:01,130 --> 00:00:06,510
Okay, so that was my, my brief overview of
what

2
00:00:06,510 --> 00:00:11,620
integration is and two or three problems
that can be used to to address.

3
00:00:11,620 --> 00:00:15,820
And I'm going to start talking about
techniques of integration.

4
00:00:17,600 --> 00:00:22,040
So these are techniques for finding
anti-derivatives of functions.

5
00:00:22,040 --> 00:00:24,380
And the first one is called integration by
parts.

6
00:00:28,910 --> 00:00:31,680
So essentially what I'm going to try and
do to get integration by

7
00:00:31,680 --> 00:00:35,820
parts is remember, when I was computing
derivatives, we had certain rules.

8
00:00:35,820 --> 00:00:37,870
One of them was the product rule which

9
00:00:37,870 --> 00:00:40,220
was the derivative of the product of two
functions.

10
00:00:41,420 --> 00:00:43,850
And so I'm just going to try and use that

11
00:00:43,850 --> 00:00:49,290
rule backwards to be able to evaluate an
anti derivative.

12
00:00:51,610 --> 00:00:55,090
So, let's start out with two functions, f
of x and g of x.

13
00:00:55,090 --> 00:00:56,960
And they need to be continuous and
integrable.

14
00:01:03,220 --> 00:01:07,550
Then if I looked at the product of their
anti derivatives,

15
00:01:10,290 --> 00:01:12,770
and here it's a little bit, a little

16
00:01:12,770 --> 00:01:15,260
bit trickier what I'm going to do with,
constant.

17
00:01:15,260 --> 00:01:17,120
Because if I said f of x, if I

18
00:01:17,120 --> 00:01:20,880
was allowed, to put in a arbitrary
constant there.

19
00:01:20,880 --> 00:01:23,150
And I did the same thing with g of x then
when

20
00:01:23,150 --> 00:01:28,368
I multiply these together I would have the
constant from f of x

21
00:01:28,368 --> 00:01:31,750
times g of x and the constant from g of x
times f

22
00:01:31,750 --> 00:01:35,660
of x, so for this to work I have to choose
my constant

23
00:01:35,660 --> 00:01:36,339
to be zero

24
00:01:40,870 --> 00:01:40,870
[SOUND].

25
00:01:40,870 --> 00:01:42,820
So I'm going to take the product of these
two

26
00:01:42,820 --> 00:01:47,220
anti derivatives, and take its derivative
using the product rule.

27
00:01:47,220 --> 00:01:48,930
And what I'm going to end up with is

28
00:01:48,930 --> 00:01:51,410
the derivative of the first function times
the second.

29
00:01:52,550 --> 00:01:54,320
So this guy here.

30
00:01:54,320 --> 00:01:57,400
Plus the derivative of the second
function.

31
00:01:57,400 --> 00:01:58,260
Times the first.

32
00:02:01,830 --> 00:02:06,890
So then if I take the integral of both
sides of that, well the

33
00:02:06,890 --> 00:02:08,890
integral sort of by the fundamental
theorem

34
00:02:08,890 --> 00:02:11,360
of calculus that's just undoing my
derivative

35
00:02:11,360 --> 00:02:17,700
operation, so I have capital F of x times
capital G of x

36
00:02:17,700 --> 00:02:23,370
is just going to be equal to the integrals
of the two terms on the right-hand side.

37
00:02:27,720 --> 00:02:29,770
And then I'm just going to rearrange these
terms

38
00:02:29,770 --> 00:02:32,410
to get something called the integration by
parts formula.

39
00:02:35,080 --> 00:02:36,020
So I'm going to take

40
00:02:38,380 --> 00:02:38,880
Oops,

41
00:02:40,950 --> 00:02:44,610
this second term, move it to the left hand
side of the equation

42
00:02:44,610 --> 00:02:47,170
and move the other two terms to the right
hand side of the equation.

43
00:02:47,170 --> 00:02:47,670
And

44
00:02:50,170 --> 00:02:52,050
now what this is telling me is that if I
can

45
00:02:52,050 --> 00:02:58,150
find, if I can break my function up into a
function

46
00:02:58,150 --> 00:03:01,970
that I can integrate, this g of x, and a
function

47
00:03:01,970 --> 00:03:05,570
that I can take its, where I can take its
derivative.

48
00:03:05,570 --> 00:03:09,730
Then I can use this formula to evaluate
this integral.

49
00:03:12,800 --> 00:03:16,240
So generally the idea is going to be to
choose.

50
00:03:16,240 --> 00:03:18,760
F of x and little g of x so that

51
00:03:18,760 --> 00:03:22,160
this final term here is something that's
easy to compute.

52
00:03:23,190 --> 00:03:29,030
So I'm starting out over here with a
capital F of x.

53
00:03:29,030 --> 00:03:31,820
So the easiest thing that I could ever
have here would just be an

54
00:03:31,820 --> 00:03:37,610
x, because if I took the derivative of
captial X, if capital, sorry capital F.

55
00:03:37,610 --> 00:03:38,270
If that's

56
00:03:38,270 --> 00:03:42,770
a constant, then I could just take this
little f,

57
00:03:42,770 --> 00:03:45,980
the derivative of a constant, sorry, if
this was x,

58
00:03:47,580 --> 00:03:49,380
little f of x is going to be a constant,

59
00:03:49,380 --> 00:03:52,380
and I can take my constant outside my
integral sign.

60
00:03:52,380 --> 00:03:56,649
And then I just have to integrate, oops,
that should be a little g of x.

61
00:03:58,350 --> 00:04:03,390
No, sorry, capital G of x never mind.
so what I'm wanting

62
00:04:03,390 --> 00:04:06,050
to do is choose these functions so that
this

63
00:04:06,050 --> 00:04:09,290
integral is going to be something that's
easy to evaluate.

64
00:04:09,290 --> 00:04:11,230
And so since I'm just stumbling over my
words

65
00:04:11,230 --> 00:04:14,120
now I probably should just skip ahead to
an example.

66
00:04:16,740 --> 00:04:19,355
So here this doesn't even look like it's
something that

67
00:04:19,355 --> 00:04:21,180
I'm going to be able to write down as a
product.

68
00:04:22,480 --> 00:04:24,330
But you always have a one.

69
00:04:26,230 --> 00:04:32,210
So if I let my capital F of x just be this
function here log of 1 plus x.

70
00:04:34,370 --> 00:04:41,410
And I let my g of x be 1 plus x, then

71
00:04:41,410 --> 00:04:44,920
my little g of x, that's just going to be
the derivative of this guy here.

72
00:04:44,920 --> 00:04:46,710
It's just going to be a constant it's
going to be one.

73
00:04:47,760 --> 00:04:54,050
So I can write this function here as
capital F of x times little g of x.

74
00:04:58,420 --> 00:05:00,900
So this was my integration by parts
formula.

75
00:05:04,540 --> 00:05:08,650
And then I just go ahead and plug these
functions

76
00:05:08,650 --> 00:05:12,530
that I've worked out here, into the
integration by parts formula.

77
00:05:12,530 --> 00:05:17,220
So, capital F of x is just log of 1 plus
x, g

78
00:05:17,220 --> 00:05:19,906
of x is equal to 1 so I didn't bother to
write that down.

79
00:05:19,906 --> 00:05:24,780
The anti derivative of that is going to be
F

80
00:05:24,780 --> 00:05:28,800
of x, capital F of x times capital G of x.

81
00:05:28,800 --> 00:05:29,540
So, that's just

82
00:05:29,540 --> 00:05:37,650
1 plus x times log of 1 plus x minus the
integral of little f of x.

83
00:05:37,650 --> 00:05:40,010
So that's going to be the derivative of
this guy here.

84
00:05:42,950 --> 00:05:46,590
Times capital g of x which is one plus x.

85
00:05:46,590 --> 00:05:50,530
So, the capital G of x ends up on top
here.

86
00:05:50,530 --> 00:05:54,070
The derivative of the logarithm is just 1
over the argument.

87
00:05:54,070 --> 00:05:56,880
So in this case, the argument is 1 plus x.

88
00:05:56,880 --> 00:06:01,870
So, little f of x is going to be 1 over 1
plus x.

89
00:06:01,870 --> 00:06:05,920
And technically I skipped a step here,
because I also have to use the chain rule.

90
00:06:05,920 --> 00:06:08,240
But the derivative of 1 plus x,

91
00:06:08,240 --> 00:06:12,753
we already said, was equal to 1.
So, it's 1 over 1 plus x, times 1.

92
00:06:14,560 --> 00:06:16,930
And now if I look at the term on the right

93
00:06:16,930 --> 00:06:20,190
here, I have 1 plus x, divided by 1 plus
x.

94
00:06:20,190 --> 00:06:22,070
So that's just going to cancel each other
out.

95
00:06:24,780 --> 00:06:28,630
So I have 1 plus x times the log of 1 plus
x.

96
00:06:28,630 --> 00:06:30,710
Minus the integral just of d x.

97
00:06:32,500 --> 00:06:33,102
And so.

98
00:06:33,102 --> 00:06:36,580
That integral is very easy to calculate, I
just end up

99
00:06:36,580 --> 00:06:40,500
with x being the anti derivative and then
because I don't have

100
00:06:40,500 --> 00:06:42,990
limits here I am asking for an
in-definitive and well just

101
00:06:42,990 --> 00:06:46,090
a anti derivative so I am adding on a
arbitrary constant C,

102
00:06:49,420 --> 00:06:51,800
so generally the bulk of the work in

103
00:06:51,800 --> 00:06:55,500
using this formula Is looking at the
original function

104
00:06:58,140 --> 00:07:03,561
and then trying to identify you know, how
am I going to write that as a capital F of

105
00:07:03,561 --> 00:07:11,270
x and a lower case g of x so I want to
split up that function.

106
00:07:11,270 --> 00:07:14,560
So that this term that I get on the
right-hand side, the term that I still

107
00:07:14,560 --> 00:07:16,700
have to integrate after I use the
integration

108
00:07:16,700 --> 00:07:20,230
by parts formula, is going to be easy to
integrate.

109
00:07:20,230 --> 00:07:23,140
And unfortunately, there's just not a
really good way to

110
00:07:23,140 --> 00:07:24,290
tell you how to do that.

111
00:07:24,290 --> 00:07:27,290
You just have to practice, and every time
you do it, you get

112
00:07:27,290 --> 00:07:30,340
a little bit better at seeing how you
could break these things up.

113
00:07:34,490 --> 00:07:37,560
So suppose I want to integrate, I want to
find

114
00:07:37,560 --> 00:07:42,250
the anti-derivative for the function x
squared times log of x.

115
00:07:46,070 --> 00:07:49,500
So this time it's going to be a little bit
simpler, I mean this is clearly.

116
00:07:50,590 --> 00:07:52,030
The product of two functions.

117
00:07:52,030 --> 00:07:54,500
I have x squared times log of x.

118
00:07:55,790 --> 00:08:00,820
So, let's just start there by saying log
of x is going to

119
00:08:00,820 --> 00:08:06,730
be the f of x and I could also choose to
make x squared the f of x.

120
00:08:06,730 --> 00:08:11,110
And log of x to be.
The, the g of x, but I'm

121
00:08:11,110 --> 00:08:12,630
going to have to integrate this thing.

122
00:08:12,630 --> 00:08:17,420
So I want to pick that in a way where it's
going to be easy to do.

123
00:08:17,420 --> 00:08:19,380
So the capital F of x, that's something

124
00:08:19,380 --> 00:08:21,050
I'm going to have to take the derivative
of.

125
00:08:22,210 --> 00:08:23,620
And it's going to be easier to take the

126
00:08:23,620 --> 00:08:26,150
derivative of this than it is to integrate
it.

127
00:08:26,150 --> 00:08:30,750
On the other hand, this is just, something
I can use my anti power rule on.

128
00:08:30,750 --> 00:08:32,790
So that's going to be something that's
probably not going to

129
00:08:32,790 --> 00:08:35,620
be too difficult to find an anti
derivative of.

130
00:08:35,620 --> 00:08:36,180
So let's

131
00:08:36,180 --> 00:08:37,890
choose that to be the part where

132
00:08:37,890 --> 00:08:39,650
I'm going to have to compute the
antiderivative.

133
00:08:40,980 --> 00:08:44,650
So, so if my my capital F of x is log of x

134
00:08:46,710 --> 00:08:49,730
and my capital G of X is 1 3rd x cubed.

135
00:08:51,010 --> 00:08:53,610
Then my little g of x is going to be x
squared.

136
00:08:56,140 --> 00:08:59,110
And so now we can figure out what the anti
derivative

137
00:08:59,110 --> 00:09:02,570
of this function is just by plugging these
values into the

138
00:09:02,570 --> 00:09:07,330
integration by parts formula, and hoping
that we're able to integrate

139
00:09:07,330 --> 00:09:09,380
the term that's going to end up on the
right-hand side.

140
00:09:11,590 --> 00:09:16,680
So, let's see.
Yeah, got it right this time.

141
00:09:16,680 --> 00:09:22,110
So the integral of capital F of x, g of x.
So, just plug everything in.

142
00:09:23,280 --> 00:09:30,510
So, capital F of x, that was the log of x.
The g of x is the x squared.

143
00:09:32,070 --> 00:09:36,710
And so my integration by parts formula
tells me that I take capital F of x times

144
00:09:36,710 --> 00:09:37,640
capital G of x.

145
00:09:37,640 --> 00:09:43,530
So I get one third x cubed times log of x
minus the integral.

146
00:09:46,750 --> 00:09:49,610
And now here I have my lowercase x, so
here

147
00:09:49,610 --> 00:09:52,740
I just have to take the derivative Of this
function

148
00:09:54,690 --> 00:09:57,710
and back in my little dictionary slide I
have the

149
00:09:57,710 --> 00:09:59,928
derivative of log of x is just 1 over x,

150
00:09:59,928 --> 00:10:06,840
so I have 1 over x times capital g of x
which is one third x cubed and so not

151
00:10:06,840 --> 00:10:09,410
only does it is it possible to integrate,
it even

152
00:10:09,410 --> 00:10:11,880
gets a little simpler since this 1 over x
is going to

153
00:10:11,880 --> 00:10:18,080
cancel one of my x's in my g of x
function.

154
00:10:18,080 --> 00:10:26,112
So I end up with one third x cubed log x
minus the integral of one third x squared.

155
00:10:29,140 --> 00:10:33,910
So I could use the anti power rule to
evaluate this guy here.

156
00:10:33,910 --> 00:10:40,320
And add up with one third x cubed times
log of x, minus one ninth x cubed.

157
00:10:40,320 --> 00:10:45,020
And then because I haven't put any limits
on this, I'm asking just for

158
00:10:45,020 --> 00:10:47,080
an anti derivative, an indefinite
integral, so

159
00:10:47,080 --> 00:10:49,000
I have to add an arbitrary constant c.

160
00:10:53,360 --> 00:10:58,040
And we're also going to be able to.
Do this for definite integrals.

161
00:10:58,040 --> 00:11:00,390
And in practice, really, it's not going to
change anything.

162
00:11:00,390 --> 00:11:03,560
You just use the trick to find the anti
derivative.

163
00:11:03,560 --> 00:11:06,370
But you can actually work though it if you
want to.

164
00:11:06,370 --> 00:11:10,220
this will probably be the one and only
time you do with the limits here.

165
00:11:14,200 --> 00:11:17,040
So this is exactly the same setup that I
had before.

166
00:11:17,040 --> 00:11:18,795
I have the.

167
00:11:18,795 --> 00:11:22,490
Derivative of capital f of x times capital
G of

168
00:11:22,490 --> 00:11:26,160
x and I'm going to integrate that over an
interval ab.

169
00:11:31,440 --> 00:11:33,530
So, because the, by the fundamental

170
00:11:33,530 --> 00:11:36,820
theoreum of calculus, this integration and
differentiation.

171
00:11:36,820 --> 00:11:39,450
They are sort of complementary operations.

172
00:11:39,450 --> 00:11:43,720
This integral is going to cancel out my
derivative and I just have to

173
00:11:43,720 --> 00:11:48,770
then evaluate my anti-derivative here at
its end points and take the difference.

174
00:11:50,630 --> 00:11:57,100
So that's going to give me capital F of b
times capital G of b minus capital F of a

175
00:11:57,100 --> 00:11:58,280
times capital G of a.

176
00:12:01,480 --> 00:12:03,440
And I can then just rearrange the terms.

177
00:12:03,440 --> 00:12:07,140
So I can use the linearity property to put
this integral sign

178
00:12:09,950 --> 00:12:11,920
in front of this function as well, so I
can write

179
00:12:11,920 --> 00:12:15,450
this integral of a sum as the sum of two
integrals.

180
00:12:17,210 --> 00:12:18,000
And that gives me.

181
00:12:19,640 --> 00:12:23,260
The integration by parts formula for a
definite integral.

182
00:12:27,760 --> 00:12:28,880
Except I believe, oh no,

183
00:12:31,700 --> 00:12:36,820
yeah, so we just end up evaluating this

184
00:12:36,820 --> 00:12:40,150
term here at it's upper and lower end
point.

185
00:12:40,150 --> 00:12:42,920
So I think this can be a little bit
confusing.

186
00:12:42,920 --> 00:12:47,130
So if you, if you're not getting the
correct answer the first time you do this.

187
00:12:47,130 --> 00:12:49,610
you might want to try and use this longer
formula.

188
00:12:49,610 --> 00:12:51,780
And see if, that helps you spot your
mistake.

189
00:12:56,570 --> 00:12:58,120
And so, one final example.

190
00:12:59,200 --> 00:13:04,340
I'll try and integrate from one to three
the function x times e to the x.

191
00:13:07,320 --> 00:13:12,235
So again I'm going to let the function
that I'm going to take the derivative of

192
00:13:12,235 --> 00:13:16,520
be x, because when I take the derivative
of that I'm just going to get one.

193
00:13:16,520 --> 00:13:18,009
That will be something easy to work with.

194
00:13:19,820 --> 00:13:28,660
The function I'm going to have to
integrate should be oh, e to the x.

195
00:13:28,660 --> 00:13:31,280
Its anti-derivative is just going to be
itself.

196
00:13:31,280 --> 00:13:34,370
So that's going to be an easy choice.

197
00:13:34,370 --> 00:13:37,390
And so I just have to then go ahead and
plug these.

198
00:13:37,390 --> 00:13:39,620
Into the integration by parts formula.

199
00:13:41,010 --> 00:13:45,930
So remember on the previous slide I'd
written this as f of

200
00:13:45,930 --> 00:13:49,480
b times g of b minus f of a times g of a.

201
00:13:49,480 --> 00:13:51,950
And that's exactly what I mean by this
notation here.

202
00:13:54,600 --> 00:13:59,350
So here what I mean is take capital F of
three times capital G of three.

203
00:13:59,350 --> 00:14:04,640
Minus capital F of 1 times capital G of 1
and the rest of

204
00:14:04,640 --> 00:14:08,840
it should be clear just because the, the
limits are still on the integral signs.

205
00:14:12,970 --> 00:14:13,890
And so I'm going to get

206
00:14:17,370 --> 00:14:20,280
this line just by plugging in.

207
00:14:20,280 --> 00:14:23,790
The functions I've worked out into this
formula here.

208
00:14:29,510 --> 00:14:32,800
And so this is my capital F and my capital
G.

209
00:14:34,250 --> 00:14:38,307
And then this one was easy to do, just
because it's its own anti-derivative.

210
00:14:42,350 --> 00:14:47,915
So this is 3 times e to the third minus

211
00:14:47,915 --> 00:14:53,900
1 times e to the 1, so this comes from the
first term here and then from this

212
00:14:53,900 --> 00:14:59,510
second term I end up with e cubed minus e.

213
00:15:01,350 --> 00:15:02,280
And I can do a little bit of

214
00:15:02,280 --> 00:15:05,630
simplification, and get the, answer 2 e
cubed.