Basic Integration
Rules
Lesson 8.1
Fitting Integrals to Basic Rules
• Consider these similar integrals
5
 x2  4 dx
5x
 x2  4 dx
5x2
 x2  4 dx
• Which one uses …
• The log rule
• The arctangent rule
• The rewrite with long division principle
Try It Out
• Decide which principle to apply …
x
 x 2  1 dx
2
  2t  1
2
4
dt
The Log Rule in Disguise
• Consider
1
dx
x
 1 e
• The quotient suggests possible Log Rule, but
the _________ is not present
• We can manipulate this to make the Log Rule
apply
• Add and subtract ex in the numerator
1  ex  ex
 1  e x dx 
The Power Rule in Disguise
• Here's another integral that doesn't seem to
fit the basic options
  cot x   ln  sin x  dx
• What are the options for u ?
• Best choice is
u  ________
du  __________________
The Power Rule in Disguise
• Then
becomes
  cot x   ln  sin x  dx
u
du

and _____________applies
• Note review of basic integration rules pg 520
• Note procedures for fitting integrands to basic
rules, pg 521
Disguises with Trig Identities
• What rules might this fit?
 tan
2
2x dx
• Note that tan2 u is ____________________
• However sec2u is on the list
• This suggests one of the
_____________________identities and we
have
Assignment
• Lesson 8.1
• Page 522
• Exercises 1 – 49 EOO
Integration by Parts
Lesson 8.2
Review Product Rule
• Recall definition of derivative of the product
of two functions
Dx  f ( x)  g ( x)  ______________________
• Now we will manipulate this to get
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  _____________________
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 = ____________
_________ = g'(x) dx
 u dv  u  v   v du
Strategy
• Given an integral  x  e x dx we split the
integrand into two parts
Note: a certain amount
• First part labeled u
• The other labeled dv
of trial and error will
happen in making this
split
• Guidelines for making the split
• The dv always includes the _______
• The ______ must be integratable
• v du is ___________________________than u dv
 u dv  u  v   v du
Making the Split
• A table to keep things organized is helpful
u
du
dv
v
 xe
x
dx
• Decide what will be the _____ and the _____
• This determines the du and the v
• Now rewrite
u  v   v du  x  e   e dx
x
x
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
•
•
•
•
•
L___________________
Inverse trigonometric
A___________
Trigonometric
E________________
Try This
5
x
 ln x dx
• Given
• Choose a u
u
du
and dv
dv
v
• Determine
the v and the du
• Substitute the values, finish integration
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
• Add the integral to both sides_____________
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
Trigonometric Integrals
Lesson 8.3
Recall Basic Identities
• Pythagorean Identities
sin 2   cos 2   1
tan 2   1  sec 2 
1  cot   csc 
2
2
• Half-Angle Formulas
1  cos 2
sin  
2
1  cos 2
2
cos  
2
2
These will be used to
integrate powers of
sin and cos
Integral of sinn x, n Odd
• Split into product of an __________________
5
4
sin
x
dx

sin

 x  sin x dx
• Make the even power a power of sin2 x
 sin
4
x  sin x dx    sin x  sin x dx
2
2
• Use the Pythagorean identity
 sin x 
2
2
sin x dx  ___________________
• Let u = cos x, du = -sin x dx
 1  u

2 2
du    1  2u  u du  ...
2
4
Integral of sinn x, n Odd
• Integrate and un-substitute
2 3 1 5
  1  2u  u du  u  u  u  C
3
5
 __________________________
2
4
• Similar strategy with cosn x, n odd
Integral of sinn x, n Even
• Use half-angle formulas
• Try
1  cos 2
sin  
2
2
4
cos
 5x dx Change to power of ________
2
1

2
cos
5
x
dx

1

cos10
x
 dx


  2 

2
• Expand the binomial, then integrate
Combinations of sin, cos
• General form
 sin
m
x  cos x dx
n
• If either n or m is odd, use techniques as
before
• Split the _____ power into an ________power
and power of one
• Use Pythagorean identity
• Specify u and du, substitute
• Usually reduces to a ____________
• Integrate, un-substitute
Combinations of sin, cos
• Consider
3
2
sin
4
x

cos
4 x dx

• Use Pythagorean identity
 sin
3
4 x  1  sin 4 x  dx    sin 4 x  sin 4 x  dx
2
• Separate and use sinn x
strategy for n odd
3
5
Combinations of tanm, secn
• When n is even
• Factor out ______________
• Rewrite remainder of integrand in terms of
Pythagorean identity sec2 x = _______________
• Then u = tan x, du = sec2x dx
• Try
sec
y

tan
y
dy

4
3
Combinations of tanm, secn
• When m is odd
• Factor out tan x sec x (for the du)
• Use identity sec2 x – 1 = tan2 x for _________
powers of tan x
• Let u = ___________________ , du = sec x tan x
• Try the same integral with this strategy
 sec
4
y  tan y dy
3
Note similar strategies for
integrals involving
combinations of
cotm x and cscn x
Integrals of
Even Powers of sec, csc
• Use the identity sec2 x – 1 = tan2 x
• Try  sec 4 3x dx 
 sec 3x  sec 3x dx 
 1  tan 3x  sec 3x dx 
  sec 3x  tan 3x  sec 3x  dx 
2
2
2
2
2
2
1 3
1
tan 3 x  tan 3 x  C
9
3
2
Wallis's Formulas
• If n is odd and (n ≥ ___) then
 /2

0
2 4 6
n 1
cos x dx      
3 5 7
n
n
• If n is even and (n ≥ ___) then
 /2

0
2 4 6
n 1 
cos x dx      

3 5 7
n 2
n
These formulas are also valid if
cosnx is replaced by _______
Wallis's Formulas
• Try it out …
 /2

0
 /2

0
sin 7 x dx
cos5 x dx
Assignment
• Lesson 8.3
• Page 540
• Exercises 1 – 41 EOO
Trigonometric
Substitution
Lesson 8.4
New Patterns for the Integrand
• Now we will look for a different set of patterns
a2  x2
a2  x2
x2  a2
• And we will use them in the context of a right
triangle
a2  x2
a
x
• Draw and label the other two triangles which
show the relationships of a and x
35
Example
• Given

dx
32  x2
θ
x 9
2
3
• Consider the labeled triangle
x
• Let x = 3 tan θ
(Why?)
• And dx = 3 sec2 θ dθ
• Then we have

3sec 2  d
9 tan 2   9
3sec2  d

 _______________________
3sec 
36
Finishing Up
• Our results are in terms of θ
• We must un-substitute back into x
ln sec  tan   C
32  x2
θ
3
x
• Use the ____________________
ln
9  x2 x
 C
3
3
37
Knowing Which Substitution
u
u
u 2  a2
38
Try It!!
• For each problem, identify which substitution
and which triangle should be used
x
3

x  9 dx
2

1 x
dx
2
x
2
4   x  1 dx
2

x 2  2 x  5 dx
39
Keep Going!
• Now finish the integration
x
3

x  9 dx
2

1 x
dx
2
x
2
4   x  1 dx
2

x  2 x  5 dx
2
40
Application
• Find the arc length of the portion of the
parabola y = 10x – x2 that is above the
x-axis
• Recall the arc length formula
b
L   1   f '( xi )  dx
2
a
41
Special Integration Formulas
• Useful formulas from Theorem 8.2
1. 
2. 
3. 
1 2
u

a  u du   a arcsin  u  a 2  u    C
2
a

1
2
2
u  a du  u  u 2  a 2  a 2  ln u  u 2  a 2  C , u  a
2
1
2
2
a  u du  u  u 2  a 2  a 2  ln u  u 2  a 2  C
2
2
2




• Look for these patterns and plug in the
a2 and u2 found in your particular integral
Assignment
• Lesson 8.4
• Page 550
• Exercises 1 – 45 EOO
Also 67, 69, 73, and 77
43
Partial Fractions
Lesson 8.5
Partial Fraction Decomposition
• Consider adding two algebraic fractions
3
2

?
x4 x5
• Partial fraction decomposition ___________
the process
x  23
3
2


2
x  x  20 x  4 x  5
Partial Fraction Decomposition
• Motivation for this process
• The separate terms are __________________
x  23
3
2
 x 2  x  20 dx   x  4 dx   x  5 dx
The Process
• Given
P( x)
f ( x) 
( x  r )n
• Where polynomial P(x) has ______________
• P(r) ≠ 0
• Then f(x) can be decomposed with this
cascading form
An
A1
A2

 ... 
2
n
x  r x  r
x  r
Strategy
Given N(x)/D(x)
1.If degree of N(x) _____________ degree of D(x)
divide the denominator into the numerator to
obtain
N ( x)
N ( x)
D( x)
  a polynomial  +
1
D( x)
Degree of N1(x) will be _________ that of D(x)
•
Now proceed with following steps for N1(x)/D(x)
Strategy
2. Factor the denominator into factors of the
form  p  x  q m and  a  x 2  b  x  c n
where a  x  b  x  c is irreducible
m
3. For each factor p  x  q  the partial fraction
must include the following sum of m
fractions
2
A
B
M

 ... 
2
m
 p  x  q  p  x  q
 p  x  q
Strategy
4. Quadratic factors: For
each
factor
of
the
n
2
form  a  x  b  x  c , the partial fraction
decomposition must include the following
sum of n fractions.
Ax  B
Cx  D
Kx  J

 ... 
2
n
2
2
2
a  x  b  x  c a  x  b  x  c
a  x  b  x  c
A Variation
• Suppose rational
function has distinct
linear factors
• Then we know
3x  1
3x  1

2
x  1  x  1   x  1
3x  1
A
B


2
x  1  x  1  x  1

A   x  1  B   x  1
 x  1 x  1
A Variation
• Now multiply through by the denominator to
clear them from the equation
3x 1  A   x  1  B   x  1
• Let x = 1 and x = -1 (Why these values?)
• Solve for A and B
What If
• Single irreducible
quadratic factor
f ( x) 
x
P( x)
2
 s x t
m
• But P(x) degree < 2m
• Then cascading form is
Am x  Bm
A1 x  B1
A2 x  B2

 ... 
2
m
2
2
2
x  s x t x  s x t
x  s x t
Gotta Try It
• Given
• Then
f ( x) 
2 x3  5
x
2
 2
2
2 x3  5
x
2
 2
2
Ax  B Cx  D
 2

2
2
x  2  x  2
...
2 x 3  5   Ax  B    x 2  2   Cx  D
Ax3  Bx 2   2 A  C  x   2 B  D 
Gotta Try It
2 x  5   Ax  B    x  2   Cx  D
3
2
Ax3  Bx 2   2 A  C  x   2 B  D 
?
• Now equate corresponding coefficients on
each side
• Solve for A, B, C, and D
2 x3  5
x
2
 2
2
Ax  B Cx  D
 2

x  2  x 2  2 2
Even More Exciting
• When
P( x)
f ( x) 
but
D( x)
• P(x) and D(x) are polynomials with
___________________________
• D(x) ≠ 0
• Example
x2  x  3
f ( x)  2
x   x  1
Combine the Methods
P( x)
• Consider f ( x) 
where
D( x)
• P(x), D(x) have no common factors
• D(x) ≠ 0
• Express as ____________functions of
Ai
x  r
n
and
x
Ak x  Bk
2
 s x t
m
Try It This Time
x2  5x  4
f ( x)  2
 x  1  x  3
• Given
x  5x  4
Ax  B
C
 2

2
 x  1  x  3  x  1 x  3
2
• Now manipulate the expression to determine
A, B, and C
Partial Fractions for Integration
• Use these principles for the following integrals
4x  3
2
  x  5 dx
3
x 1
 x 2  4 x  3 dx
Why Are We Doing This?
• Remember, the whole idea is to
make the rational function easier to integrate
x  5x  4
Ax  B
C
 2

2
 x  1  x  3  x  1 x  3
2
2x 1
1
 2

x 1 x  3
2x
1
1
 x 2  1  x 2  1  x  3 dx
Assignment
• Lesson 8.5
• Page 559
• Exercises 1 – 45 EOO
Integration by Tables
Lesson 7.1
Tables of Integrals
• Text has covered only limited variety of
integrals
• Applications in real life encounter many other
types
• _______________________to memorize all types
• Tables of integrals have been established
• Text includes list in Appendix B, pg A-18
General Table Classifications
•
•
•
•
•
•
•
•
Elementary forms
Forms involving a  b  u a  b  u  c  u
Forms involving
a  b u
u a
Forms involving
u2  a2
a u
Trigonometric forms
Inverse trigonometric forms
Exponential, logarithmic forms
Hyperbolic forms
2
2
2
2
2
, b2  4  a  c
Finding the Right Form
• For each integral
• Determine the classification
• Use the given pattern to complete the integral
5 x dx
 3  7x
 sin 5 x  sin 2 x dx
3
2 2
  25  4x  dx
4
x
  ln 2 x dx
2
1
x

tan
x dx

Reduction Formulas
• Some integral patterns in the tables have the
form
 f ( x) dx  g ( x)   h( x) dx
• This reduces a given integral to the sum of a
______________ and a ______________integral
• Given

4  3x
dx
2x
• Use formula 19
first of all
Reduction Formulas
• This gives you
1
4  3x
dx  _______________________

2
x
• Now use formula 17
du
1
 u a  b  u  a ln
a  b u  a
C
a  b u  a
and finish the integration
Assignment
• Lesson 8.6
• Page 565
• Exercises 1 – 49 EOO
Indeterminate Forms and
L’Hopital’s Rule
Lesson 8.7
Problem
• There are times when we need to evaluate
functions which are rational
x3  27
f ( x)  2
x 9
• At a specific point it may evaluate to an
indeterminate form
0
0


___

0
___
Example of the Problem
• Consider the following limit:
x3  27
lim 2
x 3 x  9
• We end up with the indeterminate form
0
0
• Note why this is indeterminate
0
 n  0n  0 n  ?
0
L’Hopital’s Rule
f ( x)
• When lim
gives
an
indeterminate
x c g ( x )
form (and the limit exists)
f '( x)
lim
x c g '( x)
• It is possible to find a limit by
• Note: this only works when the original limit
gives an ________________ form
0
0




1
0
0
Example
• Consider
lim x2  x  x
x 

As it stands this could be
• Must change to lim f '( x)
x c g '( x)
format
• So we manipulate algebraically and proceed
lim x  x  x  ________________________
2
x 
Example
• Consider
1  cos x
lim
x 0
sec x
• Why is this not a candidate for l’Hospital’s
rule?
1  cos x
lim

x 0
sec x
Example
• Try
1  cos x
lim
2
x 0
x
• When we apply l’Hospital’s
rule we get
sin x
lim
x 0 2 x
• We must apply the rule a _____________
Hints
• Manipulate the expression until you get one
of the forms
0
0




1
0
0

0
• Express the function as a _________ to get
f ( x)
g ( x)
0
Assignment
• Lesson 8.7
• Page 574
• Exercises 1 – 57 EOO
Improper Integrals
Lesson 7.7
Improper Integrals
• Note the graph of y = x -2
• We seek the area
under the curve to the
right of x = 1
______
• Thus the integral is

1
1
dx
2
x
• Known as an improper integral
To Infinity and Beyond
• To solve we write as
a limit (if the limit exists)

1
lim
1 x2 dx  ________
___

1
1
dx
2
x
Improper Integrals
• Evaluating
b
1
lim  2 dx
b 
x
1
Take the integral
 1b
lim   
b 
 x 1
 1 
lim    1  1
b 
 b 
Apply the limit
______________________
To Limit Or Not to Limit
• The limit may not exist

• Consider

1
1
dx
x
• Rewrite as a limit
and evaluate
b
1
lim  dx
b 
x
1
lim  ln | x | 
b 
b
1
_________________
To Converge Or Not

• For
1
dx
1 x p
• A limit exists (the proper integral converges)
• for _______________
• The integral _________________
• for p ≤ 1
Improper Integral to 4
• Try this one
dx
 x 2  1
• Rewrite as a limit, integrate

When f(x) Unbounded at x = c
• When vertical asymptote exists at x = c
• Given
1
x
0 1  x 2 dx
• As before, set a
limit and evaluate
t
x
• In this case the
lim 
dx  ________________
t 1 1  x 2
limit is __________
0
Using L'Hopital's Rule

• Consider
x
1

x

e
   dx
1
• Start with integration by parts
• dv _______ and u = ______________
x
x
x
1

x

e
dx


e

1

x

e





 dx
 x  e x  C
• Now apply the definition of an improper integral
Using L'Hopital's Rule
• We have

b
x
1 1  x   e dx  lim
b   e x 
 1
x
b

  lim b
 b  e
 1

 e
• Now use _______________________for the
first term
Assignment
• Lesson 8.8
• Page 585
• Exercises 1 – 61 EOO