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. . . . . .
Section5.5IntegrationbySubstitution
V63.0121, CalculusI
April28, 2009
Announcements
I Quiz6thisweekcovering5.1–5.2I Practicefinalsonthewebsite. SolutionsFriday
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. . . . . .
Outline
Announcements
LastTime: TheFundamentalTheorem(s)ofCalculus
SubstitutionforIndefiniteIntegralsTheoryExamples
SubstitutionforDefiniteIntegralsTheoryExamples
. . . . . .
OfficeHoursandotherhelp
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. . . . . .
Finalstuff
I FinalisMay8, 2:00–3:50pmin19W4101/102I Oldfinalsonline, includingFall2008I Reviewsessions: May5and6, 6:00–8:00pm, SILV 703
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Outline
Announcements
LastTime: TheFundamentalTheorem(s)ofCalculus
SubstitutionforIndefiniteIntegralsTheoryExamples
SubstitutionforDefiniteIntegralsTheoryExamples
. . . . . .
DifferentiationandIntegrationasreverseprocesses
Theorem(TheFundamentalTheoremofCalculus)
1. Let f becontinuouson [a,b]. Then
ddx
∫ x
af(t)dt = f(x)
2. Let f becontinuouson [a,b] and f = F′ forsomeotherfunction F. Then ∫ b
af(x)dx = F(b) − F(a).
. . . . . .
Techniquesofantidifferentiation?
Sofarweknowonlyafewrulesforantidifferentiation. Somearegeneral, like∫
[f(x) + g(x)] dx =
∫f(x)dx +
∫g(x)dx
Someareprettyparticular, like∫1
x√x2 − 1
dx = arcsec x + C.
Whatarewesupposedtodowiththat?
. . . . . .
Techniquesofantidifferentiation?
Sofarweknowonlyafewrulesforantidifferentiation. Somearegeneral, like∫
[f(x) + g(x)] dx =
∫f(x)dx +
∫g(x)dx
Someareprettyparticular, like∫1
x√x2 − 1
dx = arcsec x + C.
Whatarewesupposedtodowiththat?
. . . . . .
Techniquesofantidifferentiation?
Sofarweknowonlyafewrulesforantidifferentiation. Somearegeneral, like∫
[f(x) + g(x)] dx =
∫f(x)dx +
∫g(x)dx
Someareprettyparticular, like∫1
x√x2 − 1
dx = arcsec x + C.
Whatarewesupposedtodowiththat?
. . . . . .
Sofarwedon’thaveanywaytofind∫2x√x2 + 1
dx
or ∫tan x dx.
Luckily, wecanbesmartandusethe“anti”versionofoneofthemostimportantrulesofdifferentiation: thechainrule.
. . . . . .
Sofarwedon’thaveanywaytofind∫2x√x2 + 1
dx
or ∫tan x dx.
Luckily, wecanbesmartandusethe“anti”versionofoneofthemostimportantrulesofdifferentiation: thechainrule.
. . . . . .
Outline
Announcements
LastTime: TheFundamentalTheorem(s)ofCalculus
SubstitutionforIndefiniteIntegralsTheoryExamples
SubstitutionforDefiniteIntegralsTheoryExamples
. . . . . .
SubstitutionforIndefiniteIntegrals
ExampleFind ∫
x√x2 + 1
dx.
SolutionStareatthislongenoughandyounoticethetheintegrandisthederivativeoftheexpression
√1 + x2.
. . . . . .
SubstitutionforIndefiniteIntegrals
ExampleFind ∫
x√x2 + 1
dx.
SolutionStareatthislongenoughandyounoticethetheintegrandisthederivativeoftheexpression
√1 + x2.
. . . . . .
Saywhat?
Solution(Moreslowly, now)Let g(x) = x2 + 1.
Then g′(x) = 2x andso
ddx
√g(x) =
1
2√g(x)
g′(x) =x√
x2 + 1
Thus ∫x√
x2 + 1dx =
∫ (ddx
√g(x)
)dx
=√g(x) + C =
√1 + x2 + C.
. . . . . .
Saywhat?
Solution(Moreslowly, now)Let g(x) = x2 + 1. Then g′(x) = 2x andso
ddx
√g(x) =
1
2√g(x)
g′(x) =x√
x2 + 1
Thus ∫x√
x2 + 1dx =
∫ (ddx
√g(x)
)dx
=√g(x) + C =
√1 + x2 + C.
. . . . . .
Saywhat?
Solution(Moreslowly, now)Let g(x) = x2 + 1. Then g′(x) = 2x andso
ddx
√g(x) =
1
2√g(x)
g′(x) =x√
x2 + 1
Thus ∫x√
x2 + 1dx =
∫ (ddx
√g(x)
)dx
=√g(x) + C =
√1 + x2 + C.
. . . . . .
Leibniziannotationwinsagain
Solution(Sametechnique, newnotation)Let u = x2 + 1.
Then du = 2x dx and√1 + x2 =
√u. Sothe
integrandbecomescompletelytransformedinto∫x√
x2 + 1dx =
∫1√u
(12du
)=
∫1
2√udu
=
∫12u
−1/2 du
=√u + C =
√1 + x2 + C.
. . . . . .
Leibniziannotationwinsagain
Solution(Sametechnique, newnotation)Let u = x2 + 1. Then du = 2x dx and
√1 + x2 =
√u.
Sotheintegrandbecomescompletelytransformedinto∫
x√x2 + 1
dx =
∫1√u
(12du
)=
∫1
2√udu
=
∫12u
−1/2 du
=√u + C =
√1 + x2 + C.
. . . . . .
Leibniziannotationwinsagain
Solution(Sametechnique, newnotation)Let u = x2 + 1. Then du = 2x dx and
√1 + x2 =
√u. Sothe
integrandbecomescompletelytransformedinto∫x√
x2 + 1dx =
∫1√u
(12du
)=
∫1
2√udu
=
∫12u
−1/2 du
=√u + C =
√1 + x2 + C.
. . . . . .
Leibniziannotationwinsagain
Solution(Sametechnique, newnotation)Let u = x2 + 1. Then du = 2x dx and
√1 + x2 =
√u. Sothe
integrandbecomescompletelytransformedinto∫x√
x2 + 1dx =
∫1√u
(12du
)=
∫1
2√udu
=
∫12u
−1/2 du
=√u + C =
√1 + x2 + C.
. . . . . .
Leibniziannotationwinsagain
Solution(Sametechnique, newnotation)Let u = x2 + 1. Then du = 2x dx and
√1 + x2 =
√u. Sothe
integrandbecomescompletelytransformedinto∫x√
x2 + 1dx =
∫1√u
(12du
)=
∫1
2√udu
=
∫12u
−1/2 du
=√u + C =
√1 + x2 + C.
. . . . . .
TheoremoftheDay
Theorem(TheSubstitutionRule)If u = g(x) isadifferentiablefunctionwhoserangeisaninterval Iand f iscontinuouson I, then∫
f(g(x))g′(x)dx =
∫f(u)du
or ∫f(u)
dudx
dx =
∫f(u)du
. . . . . .
A polynomialexample
Example
Usethesubstitution u = x2 + 3 tofind∫
(x2 + 3)34x dx.
SolutionIf u = x2 + 3, then du = 2x dx, and 4x dx = 2du. So∫
(x2 + 3)34x dx =
∫u3 2du = 2
∫u3 du
=12u4 =
12(x2 + 3)4
. . . . . .
A polynomialexample
Example
Usethesubstitution u = x2 + 3 tofind∫
(x2 + 3)34x dx.
SolutionIf u = x2 + 3, then du = 2x dx, and 4x dx = 2du. So∫
(x2 + 3)34x dx =
∫u3 2du = 2
∫u3 du
=12u4 =
12(x2 + 3)4
. . . . . .
A polynomialexample, thehardway
Comparethistomultiplyingitout:∫(x2 + 3)34x dx =
∫ (x6 + 9x4 + 27x2 + 27
)4x dx
=
∫ (4x7 + 36x5 + 108x3 + 108x
)dx
=12x8 + 6x6 + 27x4 + 54x2
. . . . . .
Compare
Wehave∫(x2 + 3)34x dx =
12(x2 + 3)4
+ C
∫(x2 + 3)34x dx =
12x8 + 6x6 + 27x4 + 54x2
+ C
Now
12
(x2 + 3)4 =12
(x8 + 12x6 + 54x4 + 108x2 + 81
)=
12x8 + 6x6 + 27x4 + 54x2 +
812
Isthisaproblem?
No, that’swhat +C means!
. . . . . .
Compare
Wehave∫(x2 + 3)34x dx =
12(x2 + 3)4 + C∫
(x2 + 3)34x dx =12x8 + 6x6 + 27x4 + 54x2 + C
Now
12
(x2 + 3)4 =12
(x8 + 12x6 + 54x4 + 108x2 + 81
)=
12x8 + 6x6 + 27x4 + 54x2 +
812
Isthisaproblem? No, that’swhat +C means!
. . . . . .
A slickexample
Example
Find∫
tan x dx.
(Hint: tan x =sin xcos x
)
SolutionLet u = cos x. Then du = − sin x dx. So∫
tan x dx =
∫sin xcos x
dx = −∫
1udu
= − ln |u| + C
= − ln | cos x| + C = ln | sec x| + C
. . . . . .
A slickexample
Example
Find∫
tan x dx. (Hint: tan x =sin xcos x
)
SolutionLet u = cos x. Then du = − sin x dx. So∫
tan x dx =
∫sin xcos x
dx = −∫
1udu
= − ln |u| + C
= − ln | cos x| + C = ln | sec x| + C
. . . . . .
A slickexample
Example
Find∫
tan x dx. (Hint: tan x =sin xcos x
)
SolutionLet u = cos x. Then du = − sin x dx.
So∫tan x dx =
∫sin xcos x
dx = −∫
1udu
= − ln |u| + C
= − ln | cos x| + C = ln | sec x| + C
. . . . . .
A slickexample
Example
Find∫
tan x dx. (Hint: tan x =sin xcos x
)
SolutionLet u = cos x. Then du = − sin x dx. So∫
tan x dx =
∫sin xcos x
dx = −∫
1udu
= − ln |u| + C
= − ln | cos x| + C = ln | sec x| + C
. . . . . .
A slickexample
Example
Find∫
tan x dx. (Hint: tan x =sin xcos x
)
SolutionLet u = cos x. Then du = − sin x dx. So∫
tan x dx =
∫sin xcos x
dx = −∫
1udu
= − ln |u| + C
= − ln | cos x| + C = ln | sec x| + C
. . . . . .
A slickexample
Example
Find∫
tan x dx. (Hint: tan x =sin xcos x
)
SolutionLet u = cos x. Then du = − sin x dx. So∫
tan x dx =
∫sin xcos x
dx = −∫
1udu
= − ln |u| + C
= − ln | cos x| + C = ln | sec x| + C
. . . . . .
Outline
Announcements
LastTime: TheFundamentalTheorem(s)ofCalculus
SubstitutionforIndefiniteIntegralsTheoryExamples
SubstitutionforDefiniteIntegralsTheoryExamples
. . . . . .
Theorem(TheSubstitutionRuleforDefiniteIntegrals)If g′ iscontinuousand f iscontinuousontherangeof u = g(x),then ∫ b
af(g(x))g′(x)dx =
∫ g(b)
g(a)f(u)du.
. . . . . .
Example
Compute∫ π
0cos2 x sin x dx.
Solution(SlowWay)
Firstcomputetheindefiniteintegral∫
cos2 x sin x dx andthen
evaluate. Let u = cos x. Then du = − sin x dx and∫cos2 x sin x dx = −
∫u2 du
= −13u
3 + C = −13 cos
3 x + C.
Therefore ∫ π
0cos2 x sin x dx = −1
3 cos3 x
∣∣π0 = 2
3 .
. . . . . .
Example
Compute∫ π
0cos2 x sin x dx.
Solution(SlowWay)
Firstcomputetheindefiniteintegral∫
cos2 x sin x dx andthen
evaluate.
Let u = cos x. Then du = − sin x dx and∫cos2 x sin x dx = −
∫u2 du
= −13u
3 + C = −13 cos
3 x + C.
Therefore ∫ π
0cos2 x sin x dx = −1
3 cos3 x
∣∣π0 = 2
3 .
. . . . . .
Example
Compute∫ π
0cos2 x sin x dx.
Solution(SlowWay)
Firstcomputetheindefiniteintegral∫
cos2 x sin x dx andthen
evaluate. Let u = cos x. Then du = − sin x dx and∫cos2 x sin x dx = −
∫u2 du
= −13u
3 + C = −13 cos
3 x + C.
Therefore ∫ π
0cos2 x sin x dx = −1
3 cos3 x
∣∣π0 = 2
3 .
. . . . . .
Solution(FastWay)Doboththesubstitutionandtheevaluationatthesametime.
Letu = cos x. Then du = − sin x dx, u(0) = 1 and u(π) = −1. So∫ π
0cos2 x sin x dx =
∫ −1
1−u2 du
=
∫ 1
−1u2 du
= 13u
3∣∣1−1 =
23
.
. . . . . .
Solution(FastWay)Doboththesubstitutionandtheevaluationatthesametime. Letu = cos x. Then du = − sin x dx, u(0) = 1 and u(π) = −1.
So∫ π
0cos2 x sin x dx =
∫ −1
1−u2 du
=
∫ 1
−1u2 du
= 13u
3∣∣1−1 =
23
.
. . . . . .
Solution(FastWay)Doboththesubstitutionandtheevaluationatthesametime. Letu = cos x. Then du = − sin x dx, u(0) = 1 and u(π) = −1. So∫ π
0cos2 x sin x dx =
∫ −1
1−u2 du
=
∫ 1
−1u2 du
= 13u
3∣∣1−1 =
23
.
. . . . . .
Anexponentialexample
Example
Find∫ ln
√8
ln√3
e2x√e2x + 1dx
SolutionLet u = e2x, so du = 2e2x dx. Wehave∫ ln
√8
ln√3
e2x√e2x + 1dx =
12
∫ 8
3
√u + 1du
Nowlet y = u + 1, dy = du. So
12
∫ 8
3
√u + 1du =
12
∫ 9
4
√y dy =
12
∫ 9
4y1/2 dy
=12· 23y3/2
∣∣∣∣94
=13
(27− 8) =193
. . . . . .
Anexponentialexample
Example
Find∫ ln
√8
ln√3
e2x√e2x + 1dx
SolutionLet u = e2x, so du = 2e2x dx. Wehave∫ ln
√8
ln√3
e2x√e2x + 1dx =
12
∫ 8
3
√u + 1du
Nowlet y = u + 1, dy = du. So
12
∫ 8
3
√u + 1du =
12
∫ 9
4
√y dy =
12
∫ 9
4y1/2 dy
=12· 23y3/2
∣∣∣∣94
=13
(27− 8) =193
. . . . . .
Anexponentialexample
Example
Find∫ ln
√8
ln√3
e2x√e2x + 1dx
SolutionLet u = e2x, so du = 2e2x dx. Wehave∫ ln
√8
ln√3
e2x√e2x + 1dx =
12
∫ 8
3
√u + 1du
Nowlet y = u + 1, dy = du. So
12
∫ 8
3
√u + 1du =
12
∫ 9
4
√y dy =
12
∫ 9
4y1/2 dy
=12· 23y3/2
∣∣∣∣94
=13
(27− 8) =193
. . . . . .
Anotherwaytoskinthatcat
Example
Find∫ ln
√8
ln√3
e2x√e2x + 1dx
SolutionLet u = e2x + 1
, sothat du = 2e2x dx. Then∫ ln√8
ln√3
e2x√e2x + 1dx =
12
∫ 9
4
√udu
=13u3/2
∣∣∣∣94
=13
(27− 8) =193
. . . . . .
Anotherwaytoskinthatcat
Example
Find∫ ln
√8
ln√3
e2x√e2x + 1dx
SolutionLet u = e2x + 1, sothat du = 2e2x dx.
Then∫ ln√8
ln√3
e2x√e2x + 1dx =
12
∫ 9
4
√udu
=13u3/2
∣∣∣∣94
=13
(27− 8) =193
. . . . . .
Anotherwaytoskinthatcat
Example
Find∫ ln
√8
ln√3
e2x√e2x + 1dx
SolutionLet u = e2x + 1, sothat du = 2e2x dx. Then∫ ln
√8
ln√3
e2x√e2x + 1dx =
12
∫ 9
4
√udu
=13u3/2
∣∣∣∣94
=13
(27− 8) =193
. . . . . .
Anotherwaytoskinthatcat
Example
Find∫ ln
√8
ln√3
e2x√e2x + 1dx
SolutionLet u = e2x + 1, sothat du = 2e2x dx. Then∫ ln
√8
ln√3
e2x√e2x + 1dx =
12
∫ 9
4
√udu
=13u3/2
∣∣∣∣94
=13
(27− 8) =193
. . . . . .
Anotherwaytoskinthatcat
Example
Find∫ ln
√8
ln√3
e2x√e2x + 1dx
SolutionLet u = e2x + 1, sothat du = 2e2x dx. Then∫ ln
√8
ln√3
e2x√e2x + 1dx =
12
∫ 9
4
√udu
=13u3/2
∣∣∣∣94
=13
(27− 8) =193
. . . . . .
A thirdskinnedcat
Example
Find∫ ln
√8
ln√3
e2x√e2x + 1dx
SolutionLet u =
√e2x + 1, sothat
u2 = e2x + 1
=⇒ 2udu = 2e2x dx
Thus ∫ ln√8
ln√3
=
∫ 3
2u · udu =
13u3
∣∣∣∣32
=193
. . . . . .
A thirdskinnedcat
Example
Find∫ ln
√8
ln√3
e2x√e2x + 1dx
SolutionLet u =
√e2x + 1, sothat
u2 = e2x + 1 =⇒ 2udu = 2e2x dx
Thus ∫ ln√8
ln√3
=
∫ 3
2u · udu =
13u3
∣∣∣∣32
=193
. . . . . .
A thirdskinnedcat
Example
Find∫ ln
√8
ln√3
e2x√e2x + 1dx
SolutionLet u =
√e2x + 1, sothat
u2 = e2x + 1 =⇒ 2udu = 2e2x dx
Thus ∫ ln√8
ln√3
=
∫ 3
2u · udu =
13u3
∣∣∣∣32
=193
. . . . . .
ExampleFind ∫ 3π/2
πcot5
(θ
6
)sec2
(θ
6
)dθ.
Beforewedivein, thinkabout:I What“easy”substitutionsmighthelp?I Whichofthetrigfunctionssuggestsasubstitution?
. . . . . .
ExampleFind ∫ 3π/2
πcot5
(θ
6
)sec2
(θ
6
)dθ.
Beforewedivein, thinkabout:I What“easy”substitutionsmighthelp?I Whichofthetrigfunctionssuggestsasubstitution?
. . . . . .
SolutionLet φ =
θ
6. Then dφ =
16dθ.
∫ 3π/2
πcot5
(θ
6
)sec2
(θ
6
)dθ = 6
∫ π/4
π/6cot5 φ sec2 φdφ
= 6∫ π/4
π/6
sec2 φdφ
tan5 φ
Nowlet u = tanφ. So du = sec2 φdφ, and
6∫ π/4
π/6
sec2 φdφ
tan5 φ= 6
∫ 1
1/√3u−5 du
= 6(−14u−4
)∣∣∣∣11/
√3
=32
[9− 1] = 12.
. . . . . .
SolutionLet φ =
θ
6. Then dφ =
16dθ.
∫ 3π/2
πcot5
(θ
6
)sec2
(θ
6
)dθ = 6
∫ π/4
π/6cot5 φ sec2 φdφ
= 6∫ π/4
π/6
sec2 φdφ
tan5 φ
Nowlet u = tanφ. So du = sec2 φdφ, and
6∫ π/4
π/6
sec2 φdφ
tan5 φ= 6
∫ 1
1/√3u−5 du
= 6(−14u−4
)∣∣∣∣11/
√3
=32
[9− 1] = 12.
. . . . . .
Graphs
. .θ
.y
.
.π
.
.3π
2
.∫ 3π/2
πcot5
(θ
6
)sec2
(θ
6
)dθ
.φ
.y
.
.π
6
.
.π
4
.∫ π/4
π/66 cot5 φ sec2 φdφ
. . . . . .
Graphs
. .φ
.y
.
.π
6
.
.π
4
.∫ π/4
π/66 cot5 φ sec2 φdφ
.u
.y
.∫ 1
1/√36u−5 du
.
.1√3
.
.1
. . . . . .
Summary: Whatdowesubstitute?
I Linearfactors (ax + b) areeasysubstitutions: u = ax + b,du = a dx
I Lookfor function/derivativepairs intheintegrand, onetomake u andonetomake du:
I xn and xn−1 (fudgethecoefficient)I sineandcosine(fudgetheminussign)I ex and exI ax and ax (fudgethecoefficient)
I√x and
1√x(fudgethefactorof 2)
I ln x and1x