Basic Integration Rules Lesson 8.1. Fitting Integrals to Basic Rules Consider these similar integrals Which one uses … The log rule The arctangent rule.

Basic IntegrationRules

Lesson 8.1

Fitting Integrals to Basic Rules

• Consider these similar integrals

• Which one uses …• The log rule• The arctangent rule• The rewrite with long division principle

2

2 2 2

5 5 5

4 4 4

x xdx dx dx

x x x

Try It Out

• Decide which principle to apply …

2 1

xdx

x

2

2

2 1 4dt

t

The Log Rule in Disguise

• Consider

• 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

1 xdx

e

1

1

x x

x

e edx

e

The Power Rule in Disguise

• Here's another integral that doesn't seem to fit the basic options

• What are the options for u ?• Best choice is

cot ln sinx x dx

________ __________________u du

The Power Rule in Disguise

• Then

becomes and _____________applies

• Note review of basic integration rules pg 520

• Note procedures for fitting integrands to basic rules, pg 521

cot ln sinx x dx u du

Disguises with Trig Identities

• What rules might this fit?

• Note that tan2 u is ____________________• However sec2u is on the list

• This suggests one of the _____________________identities and we have

2tan 2x dx

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

• Now we will manipulate this to get

( ) ( ) ______________________xD f x g x

( ) '( ) _________________ ( ) '( )f x g x g x f x

Manipulating the Product Rule

• Now take the integral of both sides

• Which term above can be simplified?

• This gives us

( ) '( ) ( ) ( ) ( ) '( )xf x g x dx D f x g x dx g x f x dx

( ) '( ) _____________________f x g x dx

Integration by Parts

• It is customary to write this using substitution• u = f(x) du = ____________• v = g(x) _________ = g'(x) dx

( ) '( ) ( ) ( ) ( ) '( )f x g x dx f x g x g x f x dx

u dv u v v du

Strategy

• Given an integral we split the integrand into two parts • First part labeled u• The other labeled dv

• Guidelines for making the split• The dv always includes the _______• The ______ must be integratable • v du is ___________________________than u dv

Note: a certain amount of trial and error will happen in making this split

Note: a certain amount of trial and error will happen in making this split

xx e dx

u dv u v v du

Making the Split

• A table to keep things organized is helpful

• Decide what will be the _____ and the _____• This determines the du and the v• Now rewrite

xx e dxu du

dv v

x xu v v du x e e 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• L___________________• Inverse trigonometric• A___________• Trigonometric• E________________

Try This

• Given• Choose a u

and dv• Determine

the v and the du • Substitute the values, finish integration

5 lnx x dx

__________________u v v du

u du

dv v

Double Trouble

• Sometimes the second integral must also be done by parts

2 sinx x dx2 cos 2 cosx x x x dx

u du

dv v

u x2 du 2x dx

dv sin x v -cos x

Going in Circles

• When we end up with the the same as we started with

• Try• Should end up with

• Add the integral to both sides_____________

v du sinxe x dx

2 sin cos sinx x xe x dx e x e x

Application

• Consider the region bounded by y = cos x, y = 0, x = 0, and x = ½ π

• What is the volume generated by rotatingthe region around the y-axis?

What is the radius?

What is the disk thickness?

What are the limits?

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

• Half-Angle Formulas

2 2

2 2

2 2

sin cos 1

tan 1 sec

1 cot csc

2

2

1 cos 2sin

21 cos 2

cos2

These will be used to integrate powers of sin and cos

These will be used to integrate powers of sin and cos

Integral of sinn x, n Odd

• Split into product of an __________________

• Make the even power a power of sin2 x

• Use the Pythagorean identity

• Let u = cos x, du = -sin x dx

5 4sin sin sinx dx x x dx

24 2sin sin sin sinx x dx x x dx

22sin sin ___________________x x dx

22 2 41 1 2 ...u du u u du

Integral of sinn x, n Odd

• Integrate and un-substitute

• Similar strategy with cosn x, n odd

2 4 3 52 11 2

3 5__________________________

u u du u u u C

Integral of sinn x, n Even

• Use half-angle formulas

• Try Change to power of ________

• Expand the binomial, then integrate

2 1 cos 2sin

2

4cos 5x dx

2

22 1cos 5 1 cos10

2x dx x dx

Combinations of sin, cos

• General form

• 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

sin cosm nx x dx

Combinations of sin, cos

• Consider

• Use Pythagorean identity

• Separate and use sinn x strategy for n odd

3 2sin 4 cos 4x x dx

3 2 3 5sin 4 1 sin 4 sin 4 sin 4x x dx x x dx

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

• Try4 3sec tany y dy

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

4 3sec tany y dyNote similar strategies for integrals involving combinations ofcotm x and cscn x

Note similar strategies for integrals involving combinations ofcotm x and cscn x

Integrals of Even Powers of sec, csc

• Use the identity sec2 x – 1 = tan2 x • Try 4sec 3x dx

2 2

2 2

2 2 2

3

sec 3 sec 3

1 tan 3 sec 3

sec 3 tan 3 sec 3

1 1tan 3 tan 3

3 9

x x dx

x x dx

x x x dx

x x C

Wallis's Formulas

• If n is odd and (n ≥ ___) then

• If n is even and (n ≥ ___) then

/2

0

/2

0

2 4 6 1cos

3 5 7

2 4 6 1cos

3 5 7 2

n

n

nx dx

n

nx dx

n

These formulas are also valid if cosnx is replaced by _______These formulas are also valid if cosnx is replaced by _______

Wallis's Formulas

• Try it out … /2

5

0

cos x dx

/27

0

sin x dx

Assignment


Trigonometric Substitution

Lesson 8.4

New Patterns for the Integrand

• Now we will look for a different set of patterns

• And we will use them in the context of a right triangle

• Draw and label the other two triangles which show the relationships of a and x

35

2 2 2 2 2 2a x a x x a

a

x

2 2a x

Example

• Given

• Consider the labeled triangle• Let x = 3 tan θ (Why?)• And dx = 3 sec2 θ dθ

• Then we have

36

2 9

dx

x 3

x

2 23 xθ

2

2

3sec

9 tan 9

d

23sec

_______________________3sec

d

Finishing Up

• Our results are in terms of θ• We must un-substitute back into x

• Use the ____________________

37

ln sec tan C 3

x

2 23 xθ

29ln

3 3

x xC

Knowing Which Substitution

38

u

u

2 2u a

Try It!!

• For each problem, identify which substitution and which triangle should be used

39

3 2 9x x dx

2

2

1 xdx

x

2 2 5x x dx

24 1x dx

Keep Going!

• Now finish the integration

40

3 2 9x x dx

2

2

1 xdx

x

2 2 5x x dx

24 1x dx

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

41

21 '( )

b

i

a

L f x dx

Special Integration Formulas• Useful formulas from Theorem 8.2

• Look for these patterns and plug in thea2 and u2 found in your particular integral

2 2 2 2

2 2 2 2 2 2 2

2 2 2 2 2 2 2

11. arcsin

2

12. ln ,

21

3. ln2

ua u du a u a u C

a

u a du u u a a u u a C u a

a u du u u a a u u a C

Assignment


Also 67, 69, 73, and 77

43

Partial Fractions

Lesson 8.5

Partial Fraction Decomposition

• Consider adding two algebraic fractions

• Partial fraction decomposition ___________ the process

3 2?

4 5x x

2

23 3 2

20 4 5

x

x x x x

Partial Fraction Decomposition

• Motivation for this process

• The separate terms are __________________

2

23 3 2

20 4 5

xdx dx dx

x x x x

The Process

• Given

• Where polynomial P(x) has ______________• P(r) ≠ 0

• Then f(x) can be decomposed with this cascading form

( )( )

( )nP x

f xx r

1 2

2 ... nn

AA A

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

Degree of N1(x) will be _________ that of D(x)• Now proceed with following steps for N1(x)/D(x)

1( )( )a polynomial +

( ) ( )

N xN x

D x D x

Strategy

2. Factor the denominator into factors of the form

where is irreducible3. For each factor the partial fraction

must include the following sum of m fractions

2 nmp x q and a x b x c

2a x b x c mp x q

2 ... m

A B M

p x q p x q p x q

Strategy

4. Quadratic factors: For each factor of the form , the partial fraction decomposition must include the following sum of n fractions.

2 na x b x c

22 2 2... n

Ax B Cx D Kx J

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

2

3 1 3 1

1 1 1

x x

x x x

2

3 1

1 1 1

1 1

1 1

x A B

x x x

A x B x

x x

A Variation

• Now multiply through by the denominator to clear them from the equation

• Let x = 1 and x = -1 (Why these values?)• Solve for A and B

3 1 1 1x A x B x

What If

• Single irreducible quadratic factor

• But P(x) degree < 2m

• Then cascading form is

2

( )( ) m

P xf x

x s x t

1 1 2 2

22 2 2... m m

m

A x BA x B A x B

x s x t x s x t x s x t

Gotta Try It

• Given

• Then

3

22

2 5( )

2

xf x

x

3

2 222 2

3 2

3 2

2 5

22 2

...

2 5 2

2 2

x Ax B Cx D

xx x

x Ax B x Cx D

Ax Bx A C x B D

Gotta Try It

• Now equate corresponding coefficients on each side

• Solve for A, B, C, and D

3 2

3 2

2 5 2

2 2

x Ax B x Cx D

Ax Bx A C x B D

?

3

2 222 2

2 5

22 2

x Ax B Cx D

xx x

Even More Exciting

• When but

• P(x) and D(x) are polynomials with ___________________________

• D(x) ≠ 0

• Example

( )( )

( )

P xf x

D x

2

2

3( )

1

x xf x

x x

Combine the Methods

• Consider where

• P(x), D(x) have no common factors• D(x) ≠ 0

• Express as ____________functions of

( )( )

( )

P xf x

D x

2andi k k

n m

A A x B

x r x s x t

Try It This Time

• Given

• Now manipulate the expression to determine A, B, and C

2

2

5 4( )

1 3

x xf x

x x

2

2 2

5 4

31 3 1

x x Ax B C

xx x x

Partial Fractions for Integration

• Use these principles for the following integrals

2

3

4 3

5

xdx

x

2

1

4 3

xdx

x x

Why Are We Doing This?

• Remember, the whole idea is tomake the rational function easier to integrate

2

2 2

2

2 2

5 4

31 3 1

2 1 1

1 32 1 1

1 1 3

x x Ax B C

xx x x

x

x xx

dxx x x

Assignment


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• Forms involving• Forms involving• Trigonometric forms• Inverse trigonometric forms• Exponential, logarithmic forms• Hyperbolic forms

2 2a u

2 2u a

a b u 2 2, 4a b u c u b a c

a b u 2 2u a

Finding the Right Form

• For each integral• Determine the classification• Use the given pattern to complete the integral

5

3 7

x dx

x

3

2 225 4x dx

sin 5 sin 2x x dx

2 1tanx x dx

4 ln 2x x dx

Reduction Formulas• Some integral patterns in the tables have the

form

• This reduces a given integral to the sum of a ______________ and a ______________integral

• Given

• Use formula 19 first of all

( ) ( ) ( )f x dx g x h x dx

4 3

2

xdx

x

Reduction Formulas

• This gives you

• Now use formula 17

and finish the integration

1 4 3_______________________

2

xdx

x

1ln

du a b u aC

u a b u a a b u a

Assignment


Indeterminate Forms and L’Hopital’s Rule

Lesson 8.7

Problem

• There are times when we need to evaluate functions which are rational

• At a specific point it may evaluate to an indeterminate form

3

2

27( )

9

xf x

x

___0___ 0

0

Example of the Problem

• Consider the following limit:

• We end up with the indeterminate form

• Note why this is indeterminate

3

23

27lim

9x

x

x

0

0

00 0 ?

0n n n

L’Hopital’s Rule

• When gives an indeterminate

form (and the limit exists)• It is possible to find a limit by

• Note: this only works when the original limit gives an ________________ form

( )lim

( )x c

f x

g x

'( )lim

'( )x c

f x

g x

001 0

0

Example

• Consider

As it stands this could be• Must change to

format• So we manipulate algebraically and proceed

2limx

x x x

'( )lim

'( )x c

f x

g x

2lim ________________________x

x x x

Example

• Consider

• Why is this not a candidate for l’Hospital’s rule?

0

1 coslim

secx

x

x

0

1 coslim

secx

x

x

Example

• Try

• When we apply l’Hospital’s rule we get

• We must apply the rule a _____________

20

1 coslimx

x

x

0

sinlim

2x

x

x

Hints

• Manipulate the expression until you get one of the forms

• Express the function as a _________ to get

0 001 0 0

0

( )

( )

f x

g x

Assignment


Improper Integrals

Lesson 7.7

Improper Integrals

• Note the graph of y = x -2

• We seek the areaunder the curve to theright of x = 1

• Thus the integral is

• Known as an improper integral

______

21

1dx

x

To Infinity and Beyond

• To solve we write as a limit (if the limit exists)

___

2 2________1 1

1 1limdx dx

x x

Improper Integrals

• Evaluating

21

1lim

1lim

1

1lim 1 1

______________________

b

b

b

b

dxx

b

x

b

Take the integral

Apply the limit

To Limit Or Not to Limit

• The limit may not exist

• Consider

• Rewrite as a limitand evaluate

1

1dxx

1

1lim

lim ln | |1

_________________

b

b

b

dxx

bx

To Converge Or Not

• For

• A limit exists (the proper integral converges)• for _______________

• The integral _________________• for p ≤ 1

1

1pdx

x

Improper Integral to -

• Try this one

• Rewrite as a limit, integrate

4

2 1

dx

x

When f(x) Unbounded at x = c

• When vertical asymptote exists at x = c

• Given

• As before, set a limit and evaluate

• In this case the limit is __________

1

20 1

xdx

x

210

lim ________________1

t

t

xdx

x

Using L'Hopital's Rule

• Consider

• Start with integration by parts• dv _______ and u = ______________

• Now apply the definition of an improper integral

1

1 xx e dx

1 1x x x

x

x e dx e x e dx

x e C

Using L'Hopital's Rule

• We have

• Now use _______________________for the first term

11

1 lim

1lim

bx

xb

bb

xx e dx

e

b

e e

Assignment


Basic Integration Rules Lesson 8.1. Fitting Integrals to Basic Rules Consider these similar integrals Which one uses … The log rule The arctangent rule.

Documents