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Lesson 29: Integration by Substition
1. Section 5.5
Integration by Substitution
V63.0121.027, Calculus I
December 10, 2009
Announcements
Final Exam: Friday 12/18, 2:00-3:50pm, Tisch UC50
Practice ïŹnals on the website. Solutions Friday
. . . . . .
2. Schedule for next week
Tuesday, 8:00am: Review session for all students with 8:00
recitations (Tuesday or Thursday) in CIWW 109
Tuesday, 9:30am: Review session for all students with 9:30
recitations (Tuesday or Thursday) in CIWW 109
OfïŹce Hours continue
Tuesday, class: review, evaluations, movie!
Friday, 2:00pm: ïŹnal in Tisch UC50
. . . . . .
3. Resurrection Policy
If your ïŹnal score beats your midterm score, we will add 10% to
its weight, and subtract 10% from the midterm weight.
.
.
Image credit: Scott Beale / Laughing Squid
. . . . . .
4. Outline
Last Time: The Fundamental Theorem(s) of Calculus
Substitution for IndeïŹnite Integrals
Theory
Examples
Substitution for DeïŹnite Integrals
Theory
Examples
. . . . . .
5. Differentiation and Integration as reverse processes
Theorem (The Fundamental Theorem of Calculus)
1. Let f be continuous on [a, b]. Then
â« x
d
f(t) dt = f(x)
dx a
2. Let f be continuous on [a, b] and f = FâČ for some other
function F. Then
â« b
f(x) dx = F(b) â F(a).
a
. . . . . .
6. Techniques of antidifferentiation?
So far we know only a few rules for antidifferentiation. Some are
general, like
â« â« â«
[f(x) + g(x)] dx = f(x) dx + g(x) dx
. . . . . .
7. Techniques of antidifferentiation?
So far we know only a few rules for antidifferentiation. Some are
general, like
â« â« â«
[f(x) + g(x)] dx = f(x) dx + g(x) dx
Some are pretty particular, like
â«
1
â dx = arcsec x + C.
x x2 â 1
. . . . . .
8. Techniques of antidifferentiation?
So far we know only a few rules for antidifferentiation. Some are
general, like
â« â« â«
[f(x) + g(x)] dx = f(x) dx + g(x) dx
Some are pretty particular, like
â«
1
â dx = arcsec x + C.
x x2 â 1
What are we supposed to do with that?
. . . . . .
10. So far we donât have any way to ïŹnd
â«
2x
â dx
x2 + 1
or â«
tan x dx.
Luckily, we can be smart and use the âantiâ version of one of the
most important rules of differentiation: the chain rule.
. . . . . .
11. Outline
Last Time: The Fundamental Theorem(s) of Calculus
Substitution for IndeïŹnite Integrals
Theory
Examples
Substitution for DeïŹnite Integrals
Theory
Examples
. . . . . .
13. Substitution for IndeïŹnite Integrals
Example
Find â«
x
â dx.
x 2+1
Solution
Stare at this long enough and you notice the the integrand is the
â
derivative of the expression 1 + x2 .
. . . . . .
19. Leibnizian notation FTW
Solution (Same technique, new notation)
â â
Let u = x2 + 1. Then du = 2x dx and 1 + x2 = u. So the
integrand becomes completely transformed into
â« â« 1 â«
â
x dx 2 du
â
1
â du
= =
x2 + 1 u 2 u
. . . . . .
20. Leibnizian notation FTW
Solution (Same technique, new notation)
â â
Let u = x2 + 1. Then du = 2x dx and 1 + x2 = u. So the
integrand becomes completely transformed into
â« â« 1 â«
â
x dx 2 du
â
1
â du
= =
x2 + 1 â« u 2 u
1 â1/2
= 2u du
. . . . . .
21. Leibnizian notation FTW
Solution (Same technique, new notation)
â â
Let u = x2 + 1. Then du = 2x dx and 1 + x2 = u. So the
integrand becomes completely transformed into
â« â« 1 â«
â
x dx 2 du
â
1
â du
= =
x2 + 1 â« u 2 u
1 â1/2
= 2u du
â â
= u+C= 1 + x2 + C.
. . . . . .
22. Useful but unsavory variation
Solution (Same technique, new notation, more idiot-proof)
â â
Let u = x2 + 1. Then du = 2x dx and 1 + x2 = u. âSolve for
dx:â
du
dx =
2x
So the integrand becomes completely transformed into
â« â« â«
x x du 1
â dx = â · = â du
x2 + 1 u 2x 2 u
â«
1 â1/2
= 2u du
â â
= u + C = 1 + x2 + C .
Mathematicians have serious issues with mixing the x and u like
this. However, I canât deny that it works.
. . . . . .
23. Theorem of the Day
Theorem (The Substitution Rule)
If u = g(x) is a differentiable function whose range is an interval I
and f is continuous on I, then
â« â«
f(g(x))gâČ (x) dx = f(u) du
That is, if F is an antiderivative for f, then
â«
f(g(x))gâČ (x) dx = F(g(x))
In Leibniz notation:
â« â«
du
f(u) dx = f(u) du
dx
. . . . . .
24. A polynomial example
Example â«
Use the substitution u = x2 + 3 to ïŹnd (x2 + 3)3 4x dx.
. . . . . .
25. A polynomial example
Example â«
Use the substitution u = x2 + 3 to ïŹnd (x2 + 3)3 4x dx.
Solution
If u = x2 + 3, then du = 2x dx, and 4x dx = 2 du. So
â« â« â«
(x + 3) 4x dx = u 2du = 2 u3 du
2 3 3
1 4 1 2
= u = (x + 3)4
2 2
. . . . . .
26. A polynomial example, by brute force
Compare this to multiplying it out:
â« â«
2 3
( 6 )
(x + 3) 4x dx = x + 9x4 + 27x2 + 27 4x dx
â«
( 7 )
= 4x + 36x5 + 108x3 + 108x dx
1 8
= x + 6x6 + 27x4 + 54x2
2
. . . . . .
27. A polynomial example, by brute force
Compare this to multiplying it out:
â« â«
2 3
( 6 )
(x + 3) 4x dx = x + 9x4 + 27x2 + 27 4x dx
â«
( 7 )
= 4x + 36x5 + 108x3 + 108x dx
1 8
= x + 6x6 + 27x4 + 54x2
2
Which would you rather do?
. . . . . .
28. A polynomial example, by brute force
Compare this to multiplying it out:
â« â«
2 3
( 6 )
(x + 3) 4x dx = x + 9x4 + 27x2 + 27 4x dx
â«
( 7 )
= 4x + 36x5 + 108x3 + 108x dx
1 8
= x + 6x6 + 27x4 + 54x2
2
Which would you rather do?
Itâs a wash for low powers
But for higher powers, itâs much easier to do substitution.
. . . . . .
29. Compare
We have the substitution method, which, when multiplied out,
gives
â«
1
(x2 + 3)3 4x dx = (x2 + 3)4
2
1( 8 )
= x + 12x6 + 54x4 + 108x2 + 81
2
1 81
= x8 + 6x6 + 27x4 + 54x2 +
2 2
and the brute force method
â«
1
(x2 + 3)3 4x dx = x8 + 6x6 + 27x4 + 54x2
2
Is this a problem?
. . . . . .
30. Compare
We have the substitution method, which, when multiplied out,
gives
â«
1
(x2 + 3)3 4x dx = (x2 + 3)4 + C
2
1( 8 )
= x + 12x6 + 54x4 + 108x2 + 81 + C
2
1 81
= x8 + 6x6 + 27x4 + 54x2 + +C
2 2
and the brute force method
â«
1
(x2 + 3)3 4x dx = x8 + 6x6 + 27x4 + 54x2 + C
2
Is this a problem? No, thatâs what +C means!
. . . . . .
32. A slick example
Example
â«
sin x
Find tan x dx. (Hint: tan x = )
cos x
. . . . . .
33. A slick example
Example
â«
sin x
Find tan x dx. (Hint: tan x = )
cos x
Solution
Let u = cos x. Then du = â sin x dx.
. . . . . .
34. A slick example
Example
â«
sin x
Find tan x dx. (Hint: tan x = )
cos x
Solution
Let u = cos x. Then du = â sin x dx. So
â« â« â«
sin x 1
tan x dx = dx = â du
cos x u
. . . . . .
35. A slick example
Example
â«
sin x
Find tan x dx. (Hint: tan x = )
cos x
Solution
Let u = cos x. Then du = â sin x dx. So
â« â« â«
sin x 1
tan x dx = dx = â du
cos x u
= â ln |u| + C
. . . . . .
36. A slick example
Example
â«
sin x
Find tan x dx. (Hint: tan x = )
cos x
Solution
Let u = cos x. Then du = â sin x dx. So
â« â« â«
sin x 1
tan x dx = dx = â du
cos x u
= â ln |u| + C
= â ln | cos x| + C = ln | sec x| + C
. . . . . .
38. Can you do it another way?
Example
â«
sin x
Find tan x dx. (Hint: tan x = )
cos x
Solution
du
Let u = sin x. Then du = cos x dx and so dx = .
cos x
. . . . . .
39. Can you do it another way?
Example
â«
sin x
Find tan x dx. (Hint: tan x = )
cos x
Solution
du
Let u = sin x. Then du = cos x dx and so dx = .
cos x
â« â« â«
sin x u du
tan x dx = dx =
cos x cos x cos x
â« â« â«
u du u du u du
= = 2
=
cos2 x 1 â sin x 1 â u2
At this point, although itâs possible to proceed, we should
probably back up and see if the other way works quicker (it
does).
. . . . . .
40. For those who really must know all
Solution (Continued, with algebra help)
â« â« â« ( )
u du 1 1 1
tan x dx = = â du
1 â u2 2 1âu 1+u
1 1
= â ln |1 â u| â ln |1 + u| + C
2 2
1 1
= ln â + C = ln â +C
(1 â u)(1 + u) 1 â u2
1
= ln + C = ln |sec x| + C
|cos x|
. . . . . .
41. Outline
Last Time: The Fundamental Theorem(s) of Calculus
Substitution for IndeïŹnite Integrals
Theory
Examples
Substitution for DeïŹnite Integrals
Theory
Examples
. . . . . .
43. Theorem (The Substitution Rule for DeïŹnite Integrals)
If gâČ is continuous and f is continuous on the range of u = g(x),
then â« â«
b g(b)
f(g(x))gâČ (x) dx = f(u) du.
a g (a )
Why the change in the limits?
The integral on the left happens in âx-landâ
The integral on the right happens in âu-landâ, so the limits
need to be u-values
To get from x to u, apply g
. . . . . .
44. Example â«
Ï
Compute cos2 x sin x dx.
0
. . . . . .
45. Example â«
Ï
Compute cos2 x sin x dx.
0
Solution (Slow Way) â«
First compute the indeïŹnite integral cos2 x sin x dx and then
evaluate.
. . . . . .
46. Example â«
Ï
Compute cos2 x sin x dx.
0
Solution (Slow Way) â«
First compute the indeïŹnite integral cos2 x sin x dx and then
evaluate. Let u = cos x. Then du = â sin x dx and
â« â«
cos x sin x dx = â u2 du
2
= â 1 u3 + C = â 1 cos3 x + C.
3 3
Therefore
â« Ï
1 Ï
1( ) 2
cos2 x sin x dx = â cos3 x =â (â1)3 â 13 = .
0 3 0 3 3
. . . . . .
50. Solution (Fast Way)
Do both the substitution and the evaluation at the same time. Let
u = cos x. Then du = â sin x dx, u(0) = 1 and u(Ï) = â1. So
â« Ï â« â1
cos2 x sin x dx = âu2 du
0 1
â« 1
= u2 du
â1
1
1 1( ) 2
= u3 = 1 â (â1) =
3 â1 3 3
The advantage to the âfast wayâ is that you completely
transform the integral into something simpler and donât have
to go back to the original x variable.
But the slow way is just as reliable.
. . . . . .
56. Another way to skin that cat
Exampleâ
â« ln 8 â
Find â e2x e2x + 1 dx
ln 3
Solution
Let u = e2x + 1
. . . . . .
57. Another way to skin that cat
Exampleâ
â« ln 8 â
Find â e2x e2x + 1 dx
ln 3
Solution
Let u = e2x + 1, so that du = 2e2x dx.
. . . . . .
58. Another way to skin that cat
Exampleâ
â« ln 8 â
Find â e2x e2x + 1 dx
ln 3
Solution
Let u = e2x + 1, so that du = 2e2x dx. Then
â« â â«
ln 8 â 1 9â
2x
â e e2x + 1 dx = u du
ln 3 2 4
. . . . . .
59. Another way to skin that cat
Exampleâ
â« ln 8 â
Find â e2x e2x + 1 dx
ln 3
Solution
Let u = e2x + 1, so that du = 2e2x dx. Then
â« â â«
ln 8 â 1 9â
2x
â e e2x + 1 dx = u du
ln 3 2 4
9
1 3/2
= u
3 4
. . . . . .
60. Another way to skin that cat
Exampleâ
â« ln 8 â
Find â e2x e2x + 1 dx
ln 3
Solution
Let u = e2x + 1, so that du = 2e2x dx. Then
â« â â«
ln 8 â 1 9â
2x
â e e2x + 1 dx = u du
ln 3 2 4
1 3/2 9
= u
3 4
1 19
= (27 â 8) =
3 3
. . . . . .
61. A third skinned cat
Exampleâ
â« ln 8 â
Find â e2x e2x + 1 dx
ln 3
Solution
â
Let u = e2x + 1, so that
u2 = e2x + 1
. . . . . .
62. A third skinned cat
Exampleâ
â« ln 8 â
Find â e2x e2x + 1 dx
ln 3
Solution
â
Let u = e2x + 1, so that
u2 = e2x + 1 =â 2u du = 2e2x dx
. . . . . .
63. A third skinned cat
Exampleâ
â« ln 8 â
Find â e2x e2x + 1 dx
ln 3
Solution
â
Let u = e2x + 1, so that
u2 = e2x + 1 =â 2u du = 2e2x dx
Thus â« â â«
ln 8 3 3
1 3 19
â = u · u du = u =
ln 3 2 3 2 3
. . . . . .
65. Example
Find â« ( ) ( )
3Ï/2
5 Ξ 2 Ξ
cot sec dΞ.
Ï 6 6
Before we dive in, think about:
What âeasyâ substitutions might help?
Which of the trig functions suggests a substitution?
. . . . . .
72. Final Thoughts
Antidifferentiation is a ânonlinearâ problem that needs
practice, intuition, and perserverance
Worksheet in recitation (also to be posted)
The whole antidifferentiation story is in Chapter 6
. . . . . .