Download Lesson 6.2 - Coweta County Schools

6.2
Antidifferentiation by
Substitution
Quick Review
1. Evaluate the definite integral.
2
 x dx
2
2
0
dy
Find
given
dx
2. y   2 dt
3. y
4. y
5. y
1 3
8
8
 x  0 
3 0 3
3
dy
x
t
 2x
2
dx
3
2
dy
2
2
 3 2 x  x 4 x  1
  2x  x
dx
dy
2
 cos  x  1
 2 x sin x 2  1
dx
 ln  tan x  dy  1  2
1

sec x  
dx  tan x 
sin x cos x




What you’ll learn about
Indefinite Integrals
 Leibniz Notation and Antiderivatives
 Substitution in Indefinite Integrals
 Substitution in Definite Integrals

Essential Question
What are some antidifferentiation techniques
and how can I use substitution to find
indefinite or definite integrals?
Indefinite Integral
The family of all antiderivatives of a function f (x) is the
indefinite integral of f with respect to x and is denoted by
 f x  dx
If F is any function that F x   f x , then
 f x  dx  F x   C
where C is an arbitrary constant called the constant of integration.
Example Evaluating an Indefinite Integral
1. Evaluate
 2 x  cos x dx.
 x  sin x  C
2
Properties of Indefinite Integrals
1.  k f x dx  k  f x dx for any constant k
2.   f x   g x  dx   f x  dx   g x  dx
Power Functions
n 1
u
3.  u n du 
 C , when n  1
n 1
1
1
4.  u du   du  ln u  C
u
Trigonometric Formulas
 sin u du   cos u  C
 cos u du  sin u  C
 sec u du  tan u  C  csc u du   cot u  C
 sec u tan u du  sec u  C
 csc u cot u du   csc u  C
2
2
Exponential and Logarithmicu Formulas
a
 a du  ln a  C
 e du  e  C
 ln u du  u ln u  u  C
ln u
u ln u  u
 log u du   ln a du  ln a  C
u
u
a
u
Example Paying Attention to the Differential
2. Let f x   x 2  1 and u  x3 .
Find each of the following antiderivatives in terms of x.
b.  f u  du
c.  f u  dx
 f x  dx
1 3
2
a.  f x  dx   x  1 dx  x  x  C
3
1 3
2
b.  f u  du   u  1 du  u  u  C
a.
3
   
1 9
1 33
3
3
 x  x C x  x C
3
3
c.
 f u  dx   u
2


 1 dx   x
  1dx
3 2
1 7
  x  1 dx  x  x  C
7
6
Example Using Substitution
2 x3
3
Let
u

x
.
3. Evaluate  x e dx.
1 u
 x e dx  3  e du
1 u
 e C
3
2 x3
1 x3
 e C
3
du
 3x 2
dx
2
du  3x dx
1
2
du  x dx
3
Example Using Substitution
4. Evaluate
 6 x 1  x dx.
2
2
2
6
x
1

x
dx
  3 1  x 2 x  dx

 3 u
12
du
2 32
 3 u   C
3

 21  x
2

3
2
C
Let u  1  x 2 .
du
 2x
dx
du  2 x dx
Pg. 337, 6.2 #1-45 odd
6.2
Antidifferentiation by
Substitution
What you’ll learn about
Indefinite Integrals
 Leibniz Notation and Antiderivatives
 Substitution in Indefinite Integrals
 Substitution in Definite Integrals

Essential Question
What are some antidifferentiation techniques
and how can I use substitution to find
indefinite or definite integrals?
Example Setting Up a Substitution with
a Trigonometric Identity
5. Evaluate
3
sin
 x dx.
2
sin
x
dx


sin

 x sin x dx
3


   1  u  du
  1  cos x sin x dx
2
2
1 3

  u  u   C
3 

1
3
  cos x  cos x  C
3
Let u  cos x.
du
  sin x
dx
du   sin x dx
Example Evaluating a Definite Integral
by Substitution
x
6. Evaluate  2
dx.
0 x 9
Let u  x 2  9.
du
 2x
dx
du  2x dx
1
du  x dx
2
2
x
1 21
0 x 2  9 dx  2 0 u du
2
1
2
 ln u  0
2

1
2
 ln x  9
2

2
0
1 5
1
  ln 5  ln 9  ln  
2
2 9
Example Setting Up a Substitution with
a Trigonometric Identity
7. Evaluate
 sin
2
2
sin
 x dx.
x dx
1 1

    cos2 x  dx
2 2

1
1
 x  sin 2 x   C
4
2
1  cos2 x 
sin x 
2
2
1 1
  cos2 x 
2 2
Example Setting Up a Substitution with
a Trigonometric Identity
8. Evaluate
 cos
2
2
cos
 x dx.
x dx
1 1

    cos2 x  dx
2 2

1
1
 x  sin 2 x   C
4
2
1  cos2 x 
cos x 
2
2
1 1
  cos2 x 
2 2
Pg. 338, 6.2 #47-69 odd