6.9.ppt integration adopted with examples from AP Classroom

Integration by Substitution
Undoing the Chain Rule
Undoing the Chain Rule
TS: Making Decisions After Reflection
TS: Making Decisions After Reflection
& Review
& Review

Objective
Objective
 To evaluate integrals using the technique
To evaluate integrals using the technique
of integration by substitution.
of integration by substitution.

Warm Up
Warm Up
What is a synonym for the term integration?
Antidifferentation
What is integration?
Integration is a process or operation that reverses differentiation.
Integration is a process or operation that reverses differentiation.
The operation of integration determines the
original function when given its derivative.

Warm Up
Warm Up
 
4
3
Find if 5 2
dy
dx y x x
 
 
2
Find if ln 3
dy
dx y x

  
3
2 3
Evaluate 4 15 +2 5 2
x x x dx
 

3 problems:

Warm Up
Warm Up
 
4
3
Find if 5 2
dy
dx y x x
 
dy
dx   
3
4 5 2
x x

3
 
2
15 2
x 

Warm Up
Warm Up
 
2
Find if ln 3
dy
dx y x

2
dy
dx x


Warm Up
Warm Up
  
3
2 3
x x x dx
 

You have already found a function whose derivative is
the expression in the integrand, so you already have an
antiderivative.
 
4
3
Find if 5 2
dy
dx y x x
 
3
dy
dx   
3
4 5 2
x x
  
2
15 2
x 
 
4
3
5 2
x x C
 

Warm Up
Warm Up
  
3
2 3
x x x dx
 

 
4
3
5 2
x x C
 
Inside function
Derivative of
inside function
Some integrals are chain rule problems in reverse.
If the derivative of the inside function is sitting elsewhere
in the integrand, then you can use a technique called
integration by substitution to evaluate the integral.

 One method for evaluating integrals involves
One method for evaluating integrals involves
untangling the chain rule.
untangling the chain rule.
 This technique is called integration by
This technique is called integration by
substitution.
substitution.
 Integration by substitution is a technique for
Integration by substitution is a technique for
finding the antiderivative of a composite
finding the antiderivative of a composite
function.
function.

 
4
Evaluate 3 3 4
x dx
 

Let 3 4
u x
 
4
3u dx 

3
du
dx 
dx
dx
3
du dx

4
3
u dx
 

4
u du 

5
5
u
C
 
5
1
5 u C
 
 
5
1
5 3 4
x C
 
Take the
derivative of u.
Substitute into
the integral.
Always express your
answer in terms of the
original variable.
Example 1

 
20
2
Evaluate 42 +4
x x dx 

2
Let 4
u x
 
 
20
42x u dx 

2
du
dx x

dx
dx
2
du x dx
  
20
21 u du 

21
21
21
u
C
 
21
u C
 
 
21
2
4
x C
 
Take the
derivative of u.
Substitute into
the integral.
Always express your
original variable.
Example 2

2 3
Evaluate 3 5
x x dx
 

3
Let 5
u x
 
 
1
2
2 3
3 5
x x dx
 

2
3
du
dx x

dx
dx
2
3
du x dx

 
1
2
2
3x u dx 

1
2 2
3
u x dx
 

1
2
u du 
 3
2
3
2
u
C
 
3
2
2
3 u C
 
 
3
2
3
2
3 5 x C
 
Take the
derivative of u.
Substitute into
the integral.
Always express your
original variable.
Example 3

  
3
2
Evaluate 2 1
x x x dx
  

2
Let u x x
 
  
3
2 1
x u dx
 

2 1
du
dx x
 
dx
dx
 
2 1
du x dx
 
   
3
2 1
u x dx
 

3
u du 

4
4
u
C
 
4
1
4 u C
 
 
4
2
1
4 x x C
 
Always express your
original variable.
Example 4

Example 5

Example 6

Example 7

Example 8

Example 9 Integration by Substitution

Example 10 Integration by Substitution

 Experiment with different choices for
Experiment with different choices for u
u
when using integration by substitution.
when using integration by substitution.
 A good choice is one whose derivative is
A good choice is one whose derivative is
expressed elsewhere in the integrand.
expressed elsewhere in the integrand.

Conclusion
Conclusion
 To integrate by substitution, select an
To integrate by substitution, select an
expression for
expression for u
u.
.
 A good choice for
A good choice for u
u is one whose
is one whose
derivative is expressed elsewhere in the
derivative is expressed elsewhere in the
integrand.
integrand.
 Next, rewrite the integral in terms of
Next, rewrite the integral in terms of u
u.
.
 Then, simplify the integral and evaluate.
Then, simplify the integral and evaluate.

6.9.ppt integration adopted with examples from AP Classroom

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