Warm-up:
Evaluate the integrals.
 x 7
1)   3e  dx
x

2)
(
x
1
3 1 x2
)dx
Warm-up:
Evaluate the integrals.
 x 7
x
1)   3e  dx  3e  7 ln x  C
x

2)
(
x
1
3 1 x2
)dx
Warm-up:
Evaluate the integrals.
 x 7
x
1)   3e  dx  3e  7 ln x  C
x

2)
( x 
1
3
2
1
)dx  2 x  sin x  C
3 1 x2
3
3
Integration by
Substitution
Section 6.3
U-Substitution
• Can be used to transform complicated
integration problems into simpler ones.
U-Substitution
• Can be used to transform complicated
integration problems into simpler ones.
* In general, there are no hard and fast rules for choosing
u, and in some problems no choice of u will work. For
those problems, we have other methods that will be
discussed later. Choosing a u that is appropriate comes
with practice and experience, but the guidelines will help
to gain a better understanding.
Guidelines for Choosing U
1) Look for a substitution that would produce an
integral expressed entirely in terms of u and
du.
2) Evaluate the integral in terms of u.
3) Replace the u, so that your final answer is in
terms of x.
Examples
1) Evaluate
 x
2) Evaluate
 sin x  9dx
2

1
50
 2 xdx
Examples
1) Evaluate
 x
2

1
50
 2 xdx
51
u
du
50
50
C

u
du 
  u  2x
51
2x
( x 2  1)51

C
51

2) Evaluate
 sin x  9dx
  sin udu
  cos u  C
u  x2 1
du  (2 x)dx
du
 dx
2x
u  x9
du  dx
More Examples
3) Evaluate
4) Evaluate
 cos 5 xdx
 1
1

x

8


3

5
dx
More Examples
3) Evaluate
 cos 5 xdx
  cos u
4) Evaluate
 1
du 1
1
  cos udu  sin u  C
5 5
5
1
 sin 5 x  C
5
1

x

8


3

5
u  5x
du  5dx
du
 dx
5
dx
1
  5  3du  3 u 5 du
u
1
u  x 8
3
1
du  dx
4
3
3
u



C
3
C
1
4
4( x  8) 4
3du  dx
3
More difficult Examples
5) Evaluate
dx
 1  25 x 2
1

2

sec

x
dx
6) Evaluate  
x

More difficult Examples
5) Evaluate
dx
 1  25 x 2
1

2

sec

x
dx
6) Evaluate  
x

u  5x
More difficult Examples
5) Evaluate
dx
1
du
 1  25 x 2   1  u 2  5
1 du
1
1
1
 

tan
u

C

tan 1 5 x  C
2
5 1 u
5
5
1

2

sec

x
dx
6) Evaluate  
x

u  5x
du  5dx
du
 dx
5
u  x
More difficult Examples
5) Evaluate
u  5x
dx
1
du
 1  25 x 2   1  u 2  5
du  5dx
1 du
1
1
1
 

tan
u

C

tan 1 5 x  C
2
5 1 u
5
5
1

2

sec

x
dx
6) Evaluate  
x

1
du
2
  dx   sec u 
x

1
1
  dx   sec 2 u  du
x

 ln x 
1

tan u  C  ln x 
du
 dx
5
u  x
du  dx
du

1

 dx
tan x  C
Even More Difficult Examples
2
sin
x cos xdx
7) Evaluate 
8) Evaluate

e
x
x
dx
Even More Difficult Examples
7) Evaluate
8) Evaluate
 sin

e
2
x cos xdx
x
x
dx
u  sin x
Even More Difficult Examples
7) Evaluate
 sin
2
u  sin x
x cos xdx
du

  u 2 cos x
cos x

2
u du
sin 3 x
u3
C

C 
3
3
8) Evaluate


x
e
e
u
x
x
dx
(2 x)du
 2 eu du  2eu  C  2e x  C
du  (cos x)dx
du
 dx
cos x
u x
du 
1
2 x
dx
(2 x )du  dx
Difficult Examples (Conclusion)
43
5
t
3

5
t
dt
9) Evaluate 
10)Evaluate
2
x
 x  1dx
Difficult Examples (Conclusion)
9) Evaluate
t
u  3  5t
4
du  25t dt
5
3  5t dt
43
5
1
1
du
3
du


u
du
  t 4u ( 
)

4

 dt
25
4
25t
4
25t
4
1 u3


 C   3 (3  5t 5 ) 3  C
25 4
100
1
3
3
10)Evaluate
2
x
 x  1dx
u  x 1
du  dx
1
2
  (u  2u  1)u du
2
5
2
3
2
1
2
  (u  2u  u )du
7
5
3
7
5
x  u 1
x 2  (u  1) 2
x 2  u 2  2u  1
3
2 2 4 2 2 2
2
4
2
2
2
 u  u  u  C  ( x  1)  ( x  1)  ( x  1) 2  C
7
5
3
7
5
3
Homework:
page 371
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