The Idea
First Example
Formal Theorem
Examples
Parameters
Integration by Substitution
Bernd Schröder
Bernd Schröder
Integration by Substitution
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Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Symbolic Integration
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Symbolic Integration
1. Simply put, symbolic integration is about undoing
derivatives.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Symbolic Integration
1. Simply put, symbolic integration is about undoing
derivatives.
2. That is, we are given a function
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Symbolic Integration
1. Simply put, symbolic integration is about undoing
derivatives.
2. That is, we are given a function, which is actually a
derivative f 0
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Symbolic Integration
1. Simply put, symbolic integration is about undoing
derivatives.
2. That is, we are given a function, which is actually a
derivative f 0 , and we are supposed to find the original
function f .
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Symbolic Integration
1. Simply put, symbolic integration is about undoing
derivatives.
2. That is, we are given a function, which is actually a
derivative f 0 , and we are supposed to find the original
function f .
3. But most functions cannot be “obviously recognized” as
derivatives of other functions.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Symbolic Integration
1. Simply put, symbolic integration is about undoing
derivatives.
2. That is, we are given a function, which is actually a
derivative f 0 , and we are supposed to find the original
function f .
3. But most functions cannot be “obviously recognized” as
derivatives of other functions.
4. “Reversing” differentiation rules provides methods to
integrate symbolically.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Symbolic Integration
1. Simply put, symbolic integration is about undoing
derivatives.
2. That is, we are given a function, which is actually a
derivative f 0 , and we are supposed to find the original
function f .
3. But most functions cannot be “obviously recognized” as
derivatives of other functions.
4. “Reversing” differentiation rules provides methods to
integrate symbolically.
Note.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Symbolic Integration
1. Simply put, symbolic integration is about undoing
derivatives.
2. That is, we are given a function, which is actually a
derivative f 0 , and we are supposed to find the original
function f .
3. But most functions cannot be “obviously recognized” as
derivatives of other functions.
4. “Reversing” differentiation rules provides methods to
integrate symbolically.
Note. Unlike differentiation, integration is not a cut-and-dried
process.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Symbolic Integration
1. Simply put, symbolic integration is about undoing
derivatives.
2. That is, we are given a function, which is actually a
derivative f 0 , and we are supposed to find the original
function f .
3. But most functions cannot be “obviously recognized” as
derivatives of other functions.
4. “Reversing” differentiation rules provides methods to
integrate symbolically.
Note. Unlike differentiation, integration is not a cut-and-dried
process. The integration methods may not give the solution
right away
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Symbolic Integration
1. Simply put, symbolic integration is about undoing
derivatives.
2. That is, we are given a function, which is actually a
derivative f 0 , and we are supposed to find the original
function f .
3. But most functions cannot be “obviously recognized” as
derivatives of other functions.
4. “Reversing” differentiation rules provides methods to
integrate symbolically.
Note. Unlike differentiation, integration is not a cut-and-dried
process. The integration methods may not give the solution
right away, it won’t be clear which method to use
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Symbolic Integration
1. Simply put, symbolic integration is about undoing
derivatives.
2. That is, we are given a function, which is actually a
derivative f 0 , and we are supposed to find the original
function f .
3. But most functions cannot be “obviously recognized” as
derivatives of other functions.
4. “Reversing” differentiation rules provides methods to
integrate symbolically.
Note. Unlike differentiation, integration is not a cut-and-dried
process. The integration methods may not give the solution
right away, it won’t be clear which method to use (first)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Symbolic Integration
1. Simply put, symbolic integration is about undoing
derivatives.
2. That is, we are given a function, which is actually a
derivative f 0 , and we are supposed to find the original
function f .
3. But most functions cannot be “obviously recognized” as
derivatives of other functions.
4. “Reversing” differentiation rules provides methods to
integrate symbolically.
Note. Unlike differentiation, integration is not a cut-and-dried
process. The integration methods may not give the solution
right away, it won’t be clear which method to use (first) and
2
there are even functions, like e−x , for which it can be proved
that all symbolic integration methods fail.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Integration by Substitution
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Integration by Substitution
1. Integration by substitution reverses the chain rule.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Integration by Substitution
1. Integration by substitution reverses the chain rule.
(f ◦ g)’(x)=f’(g(x))g’(x)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Integration by Substitution
1. Integration by substitution reverses the chain rule.
(f ◦ g)’(x)=f’(g(x))g’(x)
2. Recognizing functions as “chain rule derivatives” sounds
like a hopeless proposition.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Integration by Substitution
1. Integration by substitution reverses the chain rule.
(f ◦ g)’(x)=f’(g(x))g’(x)
2. Recognizing functions as “chain rule derivatives” sounds
like a hopeless proposition.
3. But compositions f g(x) are easier to detect.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Integration by Substitution
1. Integration by substitution reverses the chain rule.
(f ◦ g)’(x)=f’(g(x))g’(x)
2. Recognizing functions as “chain rule derivatives” sounds
like a hopeless proposition.
3. But compositions f g(x) are easier to detect.
4. If we now substitute a variable, say, u, for the inner
function g(x), then the problem should become easier.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Integration by Substitution
1. Integration by substitution reverses the chain rule.
(f ◦ g)’(x)=f’(g(x))g’(x)
2. Recognizing functions as “chain rule derivatives” sounds
like a hopeless proposition.
3. But compositions f g(x) are easier to detect.
4. If we now substitute a variable, say, u, for the inner
function g(x), then the problem should become easier.
The details are best demonstrated with an example.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Z
Compute the Indefinite Integral
Bernd Schröder
Integration by Substitution
sin(4x) dx
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Z
Compute the Indefinite Integral
sin(4x) dx
Let u := 4x.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Z
Compute the Indefinite Integral
sin(4x) dx
Let u := 4x.
du
Then
= 4.
dx
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Z
Compute the Indefinite Integral
sin(4x) dx
Let u := 4x.
du
Then
= 4.
dx
du
So dx = .
4
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Z
Compute the Indefinite Integral
Let u := 4x.
du
Then
= 4.
dx
du
So dx = .
4
Bernd Schröder
Integration by Substitution
sin(4x) dx
Z
sin(4x) dx
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Z
Compute the Indefinite Integral
Let u := 4x.
du
Then
= 4.
dx
du
So dx = .
4
Bernd Schröder
Integration by Substitution
Z
sin(4x) dx
Z
sin(4x) dx =
sin(u) dx
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Z
Compute the Indefinite Integral
Let u := 4x.
du
Then
= 4.
dx
du
So dx = .
4
Bernd Schröder
Integration by Substitution
Z
sin(4x) dx
Z
sin(4x) dx =
sin(u)
du
4
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Z
Compute the Indefinite Integral
Let u := 4x.
du
Then
= 4.
dx
du
So dx = .
4
Bernd Schröder
Integration by Substitution
Z
sin(4x) dx
Z
sin(4x) dx =
1
=
4
sin(u)
du
4
Z
sin(u) du
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Z
Compute the Indefinite Integral
Let u := 4x.
du
Then
= 4.
dx
du
So dx = .
4
Bernd Schröder
Integration by Substitution
Z
sin(4x) dx
Z
sin(4x) dx =
sin(u)
du
4
1
=
sin(u) du
4
1
=
− cos(u) + c
4
Z
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Z
Compute the Indefinite Integral
Let u := 4x.
du
Then
= 4.
dx
du
So dx = .
4
Bernd Schröder
Integration by Substitution
Z
sin(4x) dx
Z
sin(4x) dx =
sin(u)
du
4
1
=
sin(u) du
4
1
=
− cos(u) + c
4
1
= − cos(4x) + c
4
Z
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
1
sin(4x) dx = − cos(4x) + c
4
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
1
sin(4x) dx = − cos(4x) + c
4
d
1
− cos(4x) + c
dx
4
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
1
sin(4x) dx = − cos(4x) + c
4
d
1
1
− cos(4x) + c
= − − 4 sin(4x)
dx
4
4
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
1
sin(4x) dx = − cos(4x) + c
4
d
1
1
− cos(4x) + c
= − − 4 sin(4x)
dx
4
4
= sin(4x)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
1
sin(4x) dx = − cos(4x) + c
4
d
1
1
− cos(4x) + c
= − − 4 sin(4x)
dx
4
4
√
= sin(4x)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Theorem.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Theorem. Integration by substitution.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Theorem. Integration by substitution. Let f be a function
defined on an interval.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Theorem. Integration by substitution. Let f be a function
defined on an interval. If F 0 = f and g is differentiable, then
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Theorem. Integration by substitution. Let f be a function
defined on an interval. If F 0 = f and g is differentiable, then
Z
f g(x) g0 (x) dx = F g(x) + c.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Theorem. Integration by substitution. Let f be a function
defined on an interval. If F 0 = f and g is differentiable, then
Z
f g(x) g0 (x) dx = F g(x) + c.
Duh.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Theorem. Integration by substitution. Let f be a function
defined on an interval. If F 0 = f and g is differentiable, then
Z
f g(x) g0 (x) dx = F g(x) + c.
Duh. What does that have to do with the
procedure on the previous slide?
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Theorem. Integration by substitution. Let f be a function
defined on an interval. If F 0 = f and g is differentiable, then
Z
f g(x) g0 (x) dx = F g(x) + c.
Duh. What does that have to do with the
procedure on the previous slide?
Z
Bernd Schröder
Integration by Substitution
f g(x) g0 (x) dx = F g(x) + c.
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Theorem. Integration by substitution. Let f be a function
defined on an interval. If F 0 = f and g is differentiable, then
Z
f g(x) g0 (x) dx = F g(x) + c.
Duh. What does that have to do with the
procedure on the previous slide?
Z
f g(x) g0 (x) dx = F g(x) + c.
|{z}
u
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Theorem. Integration by substitution. Let f be a function
defined on an interval. If F 0 = f and g is differentiable, then
Z
f g(x) g0 (x) dx = F g(x) + c.
Duh. What does that have to do with the
procedure on the previous slide?
Z
f g(x) g0 (x) dx = F g(x) + c.
|{z}
|{z}
u
Bernd Schröder
Integration by Substitution
u
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Theorem. Integration by substitution. Let f be a function
defined on an interval. If F 0 = f and g is differentiable, then
Z
f g(x) g0 (x) dx = F g(x) + c.
Duh. What does that have to do with the
procedure on the previous slide?
Z
f g(x) g0 (x) dx = F g(x) + c.
|{z} | {z }
|{z}
u
Bernd Schröder
Integration by Substitution
du
u
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Theorem. Integration by substitution. Let f be a function
defined on an interval. If F 0 = f and g is differentiable, then
Z
f g(x) g0 (x) dx = F g(x) + c.
Duh. What does that have to do with the
procedure on the previous slide?
Z
f g(x) g0 (x) dx = F g(x) + c.
|{z} | {z }
|{z}
u
u
du
||
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Theorem. Integration by substitution. Let f be a function
defined on an interval. If F 0 = f and g is differentiable, then
Z
f g(x) g0 (x) dx = F g(x) + c.
Duh. What does that have to do with the
procedure on the previous slide?
Z
f g(x) g0 (x) dx = F g(x) + c.
|{z} | {z }
|{z}
u
u
du
||
Z
f (u) du
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Theorem. Integration by substitution. Let f be a function
defined on an interval. If F 0 = f and g is differentiable, then
Z
f g(x) g0 (x) dx = F g(x) + c.
Duh. What does that have to do with the
procedure on the previous slide?
Z
f g(x) g0 (x) dx = F g(x) + c.
|{z} | {z }
|{z}
u
u
du
||
Z
f (u) du = F(u) + c
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why Does Integration By Substitution Work?
Theorem. Integration by substitution. Let f be a function
defined on an interval. If F 0 = f and g is differentiable, then
Z
f g(x) g0 (x) dx = F g(x) + c.
Duh. What does that have to do with the
procedure on the previous slide?
Z
f g(x) g0 (x) dx = F g(x) + c.
|{z} | {z }
|{z}
u
u
du
||
||
Z
f (u) du = F(u) + c
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the integral
Bernd Schröder
Integration by Substitution
Examples
Parameters
3
x2ex dx
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the integral
Examples
Parameters
3
x2ex dx
u := x3
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the integral
Examples
Parameters
3
x2ex dx
u := x3
du
= 3x2
dx
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the integral
Examples
Parameters
3
x2ex dx
u := x3
du
= 3x2
dx
du
dx =
3x2
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the integral
Z
Examples
Parameters
3
x2ex dx
3
x2 ex dx
u := x3
du
= 3x2
dx
du
dx =
3x2
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the integral
Z
2 x3
Examples
Parameters
3
x2ex dx
Z
x e dx =
x2 eu
du
3x2
u := x3
du
= 3x2
dx
du
dx =
3x2
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the integral
Z
2 x3
Bernd Schröder
Integration by Substitution
Parameters
3
x2ex dx
du
3x2
Z
du
=
eu
3
Z
x e dx =
u := x3
du
= 3x2
dx
du
dx =
3x2
Examples
x2 eu
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the integral
Z
2 x3
Bernd Schröder
Integration by Substitution
Parameters
3
x2ex dx
du
3x2
Z
du
=
eu
3
Z
1
=
eu du
3
Z
x e dx =
u := x3
du
= 3x2
dx
du
dx =
3x2
Examples
x2 eu
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the integral
Z
2 x3
Bernd Schröder
Integration by Substitution
Parameters
3
x2ex dx
du
3x2
Z
du
=
eu
3
Z
1
=
eu du
3
1 u
=
e +c
3
Z
x e dx =
u := x3
du
= 3x2
dx
du
dx =
3x2
Examples
x2 eu
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the integral
Z
2 x3
Bernd Schröder
Integration by Substitution
Parameters
3
x2ex dx
du
3x2
Z
du
=
eu
3
Z
1
=
eu du
3
1 u
=
e +c
3
1 x3
=
e +c
3
Z
x e dx =
u := x3
du
= 3x2
dx
du
dx =
3x2
Examples
x2 eu
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
1 3
3
x2ex dx = ex + c
3
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
1 3
3
x2ex dx = ex + c
3
d
dx
Bernd Schröder
Integration by Substitution
1 x3
e +c
3
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Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
1 3
3
x2ex dx = ex + c
3
d
dx
Bernd Schröder
Integration by Substitution
1 x3
1 2 x3 e +c
=
3x e
3
3
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
1 3
3
x2ex dx = ex + c
3
d
dx
1 x3
1 2 x3 e +c
=
3x e
3
3
= x2 ex
Bernd Schröder
Integration by Substitution
3
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
1 3
3
x2ex dx = ex + c
3
d
dx
Bernd Schröder
Integration by Substitution
1 x3
1 2 x3 e +c
=
3x e
3
3
√
3
= x2 ex
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Bernd Schröder
Integration by Substitution
Examples
Parameters
y
dy.
y2 + 1
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Examples
Parameters
y
dy.
y2 + 1
u := y2 + 1
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Examples
Parameters
y
dy.
y2 + 1
u := y2 + 1
du
= 2y
dy
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Examples
Parameters
y
dy.
y2 + 1
u := y2 + 1
du
= 2y
dy
du
dy =
2y
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Z
y
y2 + 1
Examples
Parameters
y
dy.
y2 + 1
dy
u := y2 + 1
du
= 2y
dy
du
dy =
2y
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Z
Examples
Parameters
y
dy.
y2 + 1
y
dy =
2
y +1
Z
y du
u 2y
u := y2 + 1
du
= 2y
dy
du
dy =
2y
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Z
u := y2 + 1
du
= 2y
dy
du
dy =
2y
Bernd Schröder
Integration by Substitution
Examples
Parameters
y
dy.
y2 + 1
y
dy =
2
y +1
y du
u 2y
Z
1 1
=
du
2 u
Z
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Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Z
u := y2 + 1
du
= 2y
dy
du
dy =
2y
Bernd Schröder
Integration by Substitution
Examples
Parameters
y
dy.
y2 + 1
y
dy =
2
y +1
y du
u 2y
Z
1 1
=
du
2 u
1
=
ln |u| + c
2
Z
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Z
u := y2 + 1
du
= 2y
dy
du
dy =
2y
Bernd Schröder
Integration by Substitution
Examples
Parameters
y
dy.
y2 + 1
y
dy =
2
y +1
y du
u 2y
Z
1 1
=
du
2 u
1
=
ln |u| + c
2
1 2
=
ln y + 1 + c
2
Z
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
y
1 2
+c
dy
=
ln
y
+
1
2
y +1
2
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
y
1 2
+c
dy
=
ln
y
+
1
2
y +1
2
d
dy
Bernd Schröder
Integration by Substitution
1 2
ln y + 1 + c
2
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
y
1 2
+c
dy
=
ln
y
+
1
2
y +1
2
d
dy
Bernd Schröder
Integration by Substitution
1 1
1 2
=
ln y + 1 + c
2y
2
2 y2 + 1
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
y
1 2
+c
dy
=
ln
y
+
1
2
y +1
2
d
dy
Bernd Schröder
Integration by Substitution
1 1
1 2
=
ln y + 1 + c
2y
2
2 y2 + 1
y
= 2
y +1
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
y
1 2
+c
dy
=
ln
y
+
1
2
y +1
2
d
dy
Bernd Schröder
Integration by Substitution
1 1
1 2
=
ln y + 1 + c
2y
2
2 y2 + 1
√
y
= 2
y +1
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Z
Compute the Integral
Bernd Schröder
Integration by Substitution
cot(x) dx
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Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Z
Compute the Integral
Bernd Schröder
Integration by Substitution
Z
cot(x) dx =
Parameters
cos(x)
dx
sin(x)
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Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Z
Compute the Integral
Z
cot(x) dx =
Parameters
cos(x)
dx
sin(x)
u := sin(x)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Z
Compute the Integral
Z
cot(x) dx =
Parameters
cos(x)
dx
sin(x)
u := sin(x)
du
= cos(x)
dx
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Z
Compute the Integral
Z
cot(x) dx =
Parameters
cos(x)
dx
sin(x)
u := sin(x)
du
= cos(x)
dx
du
dx =
cos(x)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Z
Compute the Integral
Z
cot(x) dx =
Parameters
cos(x)
dx
sin(x)
Z
cot(x) dx =
u := sin(x)
du
= cos(x)
dx
du
dx =
cos(x)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Z
Compute the Integral
Z
cot(x) dx =
Z
Z
cot(x) dx =
Parameters
cos(x)
dx
sin(x)
cos(x)
dx
sin(x)
u := sin(x)
du
= cos(x)
dx
du
dx =
cos(x)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Z
Compute the Integral
Z
Z
cot(x) dx =
Parameters
cos(x)
dx
sin(x)
cos(x)
dx
sin(x)
Z
cos(x) du
=
u cos(x)
Z
cot(x) dx =
u := sin(x)
du
= cos(x)
dx
du
dx =
cos(x)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Z
Compute the Integral
Z
Z
cot(x) dx =
Parameters
cos(x)
dx
sin(x)
cos(x)
dx
sin(x)
Z
cos(x) du
=
u cos(x)
Z
1
=
du
u
Z
cot(x) dx =
u := sin(x)
du
= cos(x)
dx
du
dx =
cos(x)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Z
Compute the Integral
Z
Z
cot(x) dx =
Parameters
cos(x)
dx
sin(x)
cos(x)
dx
sin(x)
Z
cos(x) du
=
u cos(x)
Z
1
=
du
u
= ln |u| + c
Z
cot(x) dx =
u := sin(x)
du
= cos(x)
dx
du
dx =
cos(x)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Z
Compute the Integral
Z
cot(x) dx =
cot(x) dx =
=
Bernd Schröder
Integration by Substitution
cos(x)
dx
sin(x)
cos(x)
dx
sin(x)
Z
cos(x) du
u cos(x)
Z
1
du
u
ln |u| + c
ln sin(x) + c
Z
Z
u := sin(x)
du
= cos(x)
dx
du
dx =
cos(x)
Parameters
=
=
=
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Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking
the Indefinite Integral
Z
cot(x) dx = ln sin(x) + c
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking
the Indefinite Integral
Z
cot(x) dx = ln sin(x) + c
d
ln sin(x) + c
dx
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking
the Indefinite Integral
Z
cot(x) dx = ln sin(x) + c
d
1
ln sin(x) + c =
cos(x)
dx
sin(x)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking
the Indefinite Integral
Z
cot(x) dx = ln sin(x) + c
d
1
ln sin(x) + c =
cos(x)
dx
sin(x)
= cot(x)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking
the Indefinite Integral
Z
cot(x) dx = ln sin(x) + c
d
1
ln sin(x) + c =
cos(x)
dx
sin(x)
√
= cot(x)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Bernd Schröder
Integration by Substitution
Examples
Parameters
sin2(x) dx
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Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
sin2 (x) =
Examples
Parameters
sin2(x) dx
1
1 − cos(2x)
2
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Examples
Parameters
sin2(x) dx
1
1 − cos(2x)
2
u := 2x
sin2 (x) =
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Examples
Parameters
sin2(x) dx
1
1 − cos(2x)
2
u := 2x
du
= 2
dx
sin2 (x) =
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Examples
Parameters
sin2(x) dx
1
1 − cos(2x)
2
u := 2x
du
= 2
dx
du
dx =
2
sin2 (x) =
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Z
Examples
Parameters
sin2(x) dx
sin2 (x) dx
1
1 − cos(2x)
2
u := 2x
du
= 2
dx
du
dx =
2
sin2 (x) =
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Z
2
Examples
Parameters
sin2(x) dx
Z
sin (x) dx =
1
1 − cos(2x) dx
2
1
1 − cos(2x)
2
u := 2x
du
= 2
dx
du
dx =
2
sin2 (x) =
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Z
2
Examples
sin2(x) dx
Z
sin (x) dx =
1
sin2 (x) =
1 − cos(2x)
2
u := 2x
du
= 2
dx
du
dx =
2
Bernd Schröder
Integration by Substitution
Parameters
1
=
2
1
1 − cos(2x) dx
2
Z
1 dx −
Z
cos(2x) dx
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Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Z
2
Examples
sin2(x) dx
Z
sin (x) dx =
1
1 − cos(2x) dx
2
1
=
1 dx − cos(2x) dx
2
Z
1
du
=
x − cos(u)
2
2
Z
1
1 − cos(2x)
2
u := 2x
du
= 2
dx
du
dx =
2
sin2 (x) =
Bernd Schröder
Integration by Substitution
Parameters
Z
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Z
2
Examples
sin2(x) dx
Z
sin (x) dx =
1
1 − cos(2x) dx
2
1
=
1 dx − cos(2x) dx
2
Z
1
du
=
x − cos(u)
2
2
1
1
=
x − sin(u) + c
2
2
Z
1
1 − cos(2x)
2
u := 2x
du
= 2
dx
du
dx =
2
sin2 (x) =
Bernd Schröder
Integration by Substitution
Parameters
Z
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Z
Examples
sin2(x) dx
Z
2
sin (x) dx =
1
1 − cos(2x) dx
2
1
1 dx − cos(2x) dx
2
Z
1
du
x − cos(u)
2
2
1
1
x − sin(u) + c
2
2
1
1
x − sin(2x) + c
2
2
Z
=
1
sin2 (x) =
1 − cos(2x)
2
u := 2x
du
= 2
dx
du
dx =
2
Bernd Schröder
Integration by Substitution
Parameters
=
=
=
Z
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite
Integral
Z
1
1
sin2(x) dx = x − sin(2x) + c
2
2
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite
Integral
Z
1
1
sin2(x) dx = x − sin(2x) + c
2
2
d 1
1
x − sin(2x) + c
dx 2
2
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite
Integral
Z
1
1
sin2(x) dx = x − sin(2x) + c
2
2
1
d 1
1
x − sin(2x) + c =
1 − cos(2x)
dx 2
2
2
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite
Integral
Z
1
1
sin2(x) dx = x − sin(2x) + c
2
2
1
d 1
1
x − sin(2x) + c =
1 − cos(2x)
dx 2
2
2
= sin2 (x)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite
Integral
Z
1
1
sin2(x) dx = x − sin(2x) + c
2
2
1
d 1
1
x − sin(2x) + c =
1 − cos(2x)
dx 2
2
2
√
= sin2 (x)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Bernd Schröder
Integration by Substitution
Examples
Parameters
2
x(x − 4) 5 dx
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Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Examples
Parameters
2
x(x − 4) 5 dx
u := x − 4
Bernd Schröder
Integration by Substitution
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Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Examples
Parameters
2
x(x − 4) 5 dx
u := x − 4
du
= 1
dx
Bernd Schröder
Integration by Substitution
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Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Examples
Parameters
2
x(x − 4) 5 dx
u := x − 4
du
= 1
dx
dx = du
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Examples
Parameters
2
x(x − 4) 5 dx
u := x − 4
du
= 1
dx
dx = du (??)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Examples
Parameters
2
x(x − 4) 5 dx
u := x − 4
du
= 1
dx
dx = du (??)
x = u+4
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Examples
Parameters
2
x(x − 4) 5 dx
u := x − 4
du
= 1
dx
dx = du (??)
x = u+4
Bernd Schröder
Integration by Substitution
(!!)
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Parameters
2
x(x − 4) 5 dx
Compute the Integral
Z
Examples
2
x(x − 4) 5 dx
u := x − 4
du
= 1
dx
dx = du (??)
x = u+4
Bernd Schröder
Integration by Substitution
(!!)
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Parameters
2
x(x − 4) 5 dx
Compute the Integral
Z
Examples
2
x(x − 4) 5 dx =
Z
2
xu 5 du
u := x − 4
du
= 1
dx
dx = du (??)
x = u+4
Bernd Schröder
Integration by Substitution
(!!)
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Parameters
2
x(x − 4) 5 dx
Compute the Integral
Z
Examples
2
x(x − 4) 5 dx =
Z
Z
=
2
xu 5 du
2
(u + 4)u 5 du
u := x − 4
du
= 1
dx
dx = du (??)
x = u+4
Bernd Schröder
Integration by Substitution
(!!)
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
2
2
x(x − 4) 5 dx =
Z
2
xu 5 du
Z
=
u := x − 4
du
= 1
dx
dx = du (??)
x = u+4
Bernd Schröder
Integration by Substitution
Parameters
x(x − 4) 5 dx
Compute the Integral
Z
Examples
Z
=
2
(u + 4)u 5 du
7
2
u 5 + 4u 5 du
(!!)
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Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
2
2
x(x − 4) 5 dx =
Z
2
xu 5 du
Z
=
u := x − 4
du
= 1
dx
dx = du (??)
x = u+4
Bernd Schröder
Integration by Substitution
Parameters
x(x − 4) 5 dx
Compute the Integral
Z
Examples
2
(u + 4)u 5 du
Z
7
2
=
u 5 + 4u 5 du
=
5 12 20 7
u 5 + u5 + c
12
7
(!!)
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Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
2
2
x(x − 4) 5 dx =
Z
2
xu 5 du
Z
=
u := x − 4
du
= 1
dx
dx = du (??)
x = u+4
Bernd Schröder
Integration by Substitution
(!!)
Parameters
x(x − 4) 5 dx
Compute the Integral
Z
Examples
Z
=
2
(u + 4)u 5 du
7
2
u 5 + 4u 5 du
5 12 20 7
u 5 + u5 + c
12
7
12
7
5
20
=
(x − 4) 5 + (x − 4) 5 + c
12
7
=
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Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
2
12
7
5
20
x(x − 4) 5 dx = (x − 4) 5 + (x − 4) 5 + c
12
7
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
2
12
7
5
20
x(x − 4) 5 dx = (x − 4) 5 + (x − 4) 5 + c
12
7
12
7
d 5
20
5
5
(x − 4) + (x − 4) + c
dx 12
7
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
2
12
7
5
20
x(x − 4) 5 dx = (x − 4) 5 + (x − 4) 5 + c
12
7
12
7
d 5
20
5
5
(x − 4) + (x − 4) + c
dx 12
7
7
2
20 7
5 12
=
(x − 4) 5 +
(x − 4) 5
12 5
7 5
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
2
12
7
5
20
x(x − 4) 5 dx = (x − 4) 5 + (x − 4) 5 + c
12
7
12
7
d 5
20
5
5
(x − 4) + (x − 4) + c
dx 12
7
7
2
20 7
5 12
=
(x − 4) 5 +
(x − 4) 5
12 5
7 5
7
2
= (x − 4) 5 + 4(x − 4) 5
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
2
12
7
5
20
x(x − 4) 5 dx = (x − 4) 5 + (x − 4) 5 + c
12
7
12
7
d 5
20
5
5
(x − 4) + (x − 4) + c
dx 12
7
7
2
20 7
5 12
=
(x − 4) 5 +
(x − 4) 5
12 5
7 5
7
2
= (x − 4) 5 + 4(x − 4) 5
2
= (x − 4) 5 (x − 4 + 4)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
2
12
7
5
20
x(x − 4) 5 dx = (x − 4) 5 + (x − 4) 5 + c
12
7
12
7
d 5
20
5
5
(x − 4) + (x − 4) + c
dx 12
7
7
2
20 7
5 12
=
(x − 4) 5 +
(x − 4) 5
12 5
7 5
7
2
= (x − 4) 5 + 4(x − 4) 5
2
= (x − 4) 5 (x − 4 + 4)
2
= (x − 4) 5 x
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Checking the Indefinite Integral
Z
2
12
7
5
20
x(x − 4) 5 dx = (x − 4) 5 + (x − 4) 5 + c
12
7
12
7
d 5
20
5
5
(x − 4) + (x − 4) + c
dx 12
7
7
2
20 7
5 12
=
(x − 4) 5 +
(x − 4) 5
12 5
7 5
7
2
= (x − 4) 5 + 4(x − 4) 5
2
= (x − 4) 5 (x − 4 + 4)
√
2
= (x − 4) 5 x
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
1. We will ultimately work with problems that involve more
than one variable.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
1. We will ultimately work with problems that involve more
than one variable. For example,
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
1. We will ultimately work with problems that involve more
than one variable. For example,
1.1 Universal gas law: pV = νRT.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
1. We will ultimately work with problems that involve more
than one variable. For example,
1.1 Universal gas law: pV = νRT. And if that’s not bad
enough
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
1. We will ultimately work with problems that involve more
than one variable. For example,
1.1 Universal gas law: pV = νRT. And if that’s not bad
the
enough,
van der Waals equation
ν 2a
p + 2 (V − νb) = νRT
V
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
1. We will ultimately work with problems that involve more
than one variable. For example,
1.1 Universal gas law: pV = νRT. And if that’s not bad
the
enough,
van der Waals equation
ν 2a
p + 2 (V − νb) = νRT is a more accurate model.
V
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
1. We will ultimately work with problems that involve more
than one variable. For example,
1.1 Universal gas law: pV = νRT. And if that’s not bad
the
enough,
van der Waals equation
ν 2a
p + 2 (V − νb) = νRT is a more accurate model.
V
1.2 Harmonic oscillators:
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
1. We will ultimately work with problems that involve more
than one variable. For example,
1.1 Universal gas law: pV = νRT. And if that’s not bad
the
enough,
van der Waals equation
ν 2a
p + 2 (V − νb) = νRT is a more accurate model.
V
r
1
k
1.2 Harmonic oscillators: Frequency f =
2π m
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
1. We will ultimately work with problems that involve more
than one variable. For example,
1.1 Universal gas law: pV = νRT. And if that’s not bad
the
enough,
van der Waals equation
ν 2a
p + 2 (V − νb) = νRT is a more accurate model.
V
r
1
k
, differential
1.2 Harmonic oscillators: Frequency f =
2π m
equation mx00 + cx0 + kx = F(t).
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
1. We will ultimately work with problems that involve more
than one variable. For example,
1.1 Universal gas law: pV = νRT. And if that’s not bad
the
enough,
van der Waals equation
ν 2a
p + 2 (V − νb) = νRT is a more accurate model.
V
r
1
k
, differential
1.2 Harmonic oscillators: Frequency f =
2π m
equation mx00 + cx0 + kx = F(t).
2. In terms of working with numbers
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
1. We will ultimately work with problems that involve more
than one variable. For example,
1.1 Universal gas law: pV = νRT. And if that’s not bad
the
enough,
van der Waals equation
ν 2a
p + 2 (V − νb) = νRT is a more accurate model.
V
r
1
k
, differential
1.2 Harmonic oscillators: Frequency f =
2π m
equation mx00 + cx0 + kx = F(t).
2. In terms of working with numbers, would you rather work
with π
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
1. We will ultimately work with problems that involve more
than one variable. For example,
1.1 Universal gas law: pV = νRT. And if that’s not bad
the
enough,
van der Waals equation
ν 2a
p + 2 (V − νb) = νRT is a more accurate model.
V
r
1
k
, differential
1.2 Harmonic oscillators: Frequency f =
2π m
equation mx00 + cx0 + kx = F(t).
2. In terms of working with numbers, would you rather work
with π or with 3.141592653589793?
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
3. Parameters make our solutions applicable to more
problems.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
3. Parameters make our solutions applicable to more
problems. For example,
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
3. Parameters make our solutions applicable to more
problems. For example,
3.1 We can describe the graph of any quadratic function
f (x) = a(x − h)2 + k.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
3. Parameters make our solutions applicable to more
problems. For example,
3.1 We can describe the graph of any quadratic function
f (x) = a(x − h)2 + k.
3.2 We can optimize the surface area of a rectangular box for
any fixed volume, rather than for one where the volume is
given by a number.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
3. Parameters make our solutions applicable to more
problems. For example,
3.1 We can describe the graph of any quadratic function
f (x) = a(x − h)2 + k.
3.2 We can optimize the surface area of a rectangular box for
any fixed volume, rather than for one where the volume is
given by a number.
4. Solving more routine problems is not the answer.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
3. Parameters make our solutions applicable to more
problems. For example,
3.1 We can describe the graph of any quadratic function
f (x) = a(x − h)2 + k.
3.2 We can optimize the surface area of a rectangular box for
any fixed volume, rather than for one where the volume is
given by a number.
4. Solving more routine problems is not the answer. (That’s
what computers are for.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
3. Parameters make our solutions applicable to more
problems. For example,
3.1 We can describe the graph of any quadratic function
f (x) = a(x − h)2 + k.
3.2 We can optimize the surface area of a rectangular box for
any fixed volume, rather than for one where the volume is
given by a number.
4. Solving more routine problems is not the answer. (That’s
what computers are for. Parametrize the problem and
program it.)
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Examples
Parameters
Why We Need to Work With Parameters
3. Parameters make our solutions applicable to more
problems. For example,
3.1 We can describe the graph of any quadratic function
f (x) = a(x − h)2 + k.
3.2 We can optimize the surface area of a rectangular box for
any fixed volume, rather than for one where the volume is
given by a number.
4. Solving more routine problems is not the answer. (That’s
what computers are for. Parametrize the problem and
program it.)
5. You can gain deeper insights by solving harder problems.
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Bernd Schröder
Integration by Substitution
Examples
Parameters
1
dx
(x − a)n
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Examples
Parameters
1
dx
(x − a)n
u := x − a
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Examples
Parameters
1
dx
(x − a)n
u := x − a
du
= 1
dx
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Examples
Parameters
1
dx
(x − a)n
u := x − a
du
= 1
dx
dx = du
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Z
Examples
Parameters
1
dx
(x − a)n
1
dx
(x − a)n
u := x − a
du
= 1
dx
dx = du
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Z
Examples
Parameters
1
dx
(x − a)n
1
dx =
(x − a)n
Z
u−n du
u := x − a
du
= 1
dx
dx = du
Bernd Schröder
Integration by Substitution
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Z
1
dx =
(x − a)n
=
Bernd Schröder
Integration by Substitution
Parameters
1
dx
(x − a)n
Z
u := x − a
du
= 1
dx
dx = du
Examples
u−n du
1 1−n
+ c;
1−n u
for n 6= 1,
ln |u| + c; for n = 1,
logo1
Louisiana Tech University, College of Engineering and Science
The Idea
First Example
Formal Theorem
Z
Compute the Integral
Z
Bernd Schröder
Integration by Substitution
Parameters
1
dx
(x − a)n
1
dx =
(x − a)n
Z
=
u−n du
1 1−n
+ c;
1−n u
1
1−n + c;
1−n (x − a)
=
u := x − a
du
= 1
dx
dx = du
Examples
for n 6= 1,
ln |u| + c; for n = 1,
for n 6= 1,
ln |x − a| + c; for n = 1.
logo1
Louisiana Tech University, College of Engineering and Science