Integration by Parts
Lesson 8.2
Review Product Rule
• Recall definition of derivative of the product
of two functions
Dx  f ( x)  g ( x)  f ( x)  g '( x)  g ( x)  f '( x)
• Now we will manipulate this to get
f ( x)  g '( x)  Dx  f ( x)  g ( x)  g ( x)  f '( x)
Manipulating the Product Rule
• Now take the integral of both sides
 f ( x)  g '( x) dx   D  f ( x)  g ( x) dx   g ( x)  f '( x) dx
x
• Which term above can be simplified?
• This gives us
 f ( x)  g '( x) dx  f ( x)  g ( x)   g ( x)  f '( x) dx
Integration by Parts
 f ( x)  g '( x) dx  f ( x)  g ( x)   g ( x)  f '( x) dx
• It is customary to write this using substitution
 u = f(x)
 v = g(x)
du = f '(x) dx
dv = g'(x) dx
 u dv  u  v   v du
Strategy
x
x

e
dx we split the
• Given an integral 
integrand into two parts
Note: a certain
 First part labeled u
 The other labeled dv
amount of trial and
error will happen in
making this split
• Guidelines for making the split
 The dv always includes the dx
 The dv must be integratable
 v du is easier to integrate than u dv
 u dv  u  v   v du
Making the Split
• A table to keep things organized is helpful
u
dv
x
ex dx
du
dx
v
ex
 xe
• Decide what will be the u and the dv
• This determines the du and the v
• Now rewrite
u  v   v du  x  e   e dx
x
x
x
dx
Strategy Hint
• Trick is to select the correct function for u
• A rule of thumb is the LIATE hierarchy rule
The u should be first available from
 Logarithmic
 Inverse trigonometric
 Algebraic
 Trigonometric
 Exponential
Try This
• Given
 x ln x dx
• Choose a u
u
du
and dv
dv
v
• Determine
the v and the du
• Substitute the values, finish integration
5
u  v   v du  __________________
Double Trouble
• Sometimes the second integral must also be
done by parts
x
2
sin x dx
u
x2
du
2x dx
dv
sin x
v
-cos x
 x cos x  2 x  cos x dx
2
u
du
dv
v
Going in Circles

• When we end up with the  v du the same
as we started with
x
e
• Try  sin x dx
• Should end up with
e
x
sin x dx  e cos x  e sin x   e sin x dx
x
x
x
• Add the integral to both sides, divide by 2
2  e sin x dx  e cos x  e sin x
x
x
x
Application
• Consider the region
bounded by y = cos x,
y = 0, x = 0, and
x=½π
• What is the volume
generated by rotating
the region around the
y-axis?
What is the radius?
What is the disk thickness?
What are the limits?
Assignment
• Lesson 8.2A
• Page 531
• Exercises 1 – 35 odd
• Lesson 8.2B
• Page 532
• Exercises 47 – 57, 99 – 105 odd