MATH 27 Lecture Guide UNIT 2

8/10/2019 MATH 27 Lecture Guide UNIT 2

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MATH27LectureGuideUNIT2albabierra

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UNIT 2. TECHNIQUES OF INTEGRATION MATH 27 LECTURE GUIDE

Objectives: By the end of the unit, a student should be able to perform integration by parts; evaluate integrals of powers of trigonometric functions; use trigonometric substitution to evaluate some integral forms; decompose rational functions to partial fractions;

use proper substitute to evaluate some integral forms; and determine and execute the proper technique in evaluating integrals

__________________________

MUST !!! REVIEW on integral forms in UNIT 1. These will be the basis of the other solvable integralforms in this unit. Also, review the derivatives for solving differentials in case substitution will be usedin solving integrals.

2.1 Integration by Parts (TC7 pp. 574-582 / TCWAG pp. 531-536)

Let u and be functions of .

Product rule for differentiation: [ ] [ ] [ ]uDvvDuvuD xxx +=

Product rule for differentials: [ ] duvdvuvud +=

[ ] += duvdvuvud += duvdvuvu = duvvudvu

MUST REMEMBER!!! Integration by parts (IBP).

An integral form ( ) dxxf can be expressed as dvu which is, in turn, equal to duvvu .Once u and dv are determined, solve du from u , and dv from . Then, solve the resultingform.

( ) = dxxf = dvu duvvu

TO DO!!! Evaluate the following.

1. xdxcosx 2. xdxsinArc


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MUST REMEMBER!!! For xdxsinm , xdxcosn and xdxcosxsin nm .

CASE 1. m or n is odd. Separate one factor of the odd-powered function. Express the rest in

terms of the other function using 122 =+ xcosxsin .Proceed withsubstitution!

CASE 2. m and n are even. Use2

212 xcosxsin = or

2

212 xcosxcos += .


1. xdxsin5 3. xdxcosxsin 22

1. xdxcosxsin 32

EXERCISE.

Evaluate the following.

1. xdxcos5 2. xdxcosxsin 35 3. xdxcos4

MUST REMEMBER!!! For xdxtanm and xdxcotn .Separate xtan2 or xcot2 . Express it in terms of sec or csc using

122 = xsecxtan or 122 = xcscxcot for even powers of tan or cot .Proceed with substitution!


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MUST REMEMBER!!! For xdxsecm and xdxcscm .CASE 1. m is even. Separate xsec2 or xcsc2 . Express the rest in terms of xtan or

cot using xtanxsec 22 1+= or xcotxcsc 22 1+= . Proceed withsubstitution!

CASE 2. m is odd. Use IBP with xsecdv 2= or xcscdv 2= .


2. xdxcot4 2. xdxtan3


1. xdxcsc4 1. xdxsec

3

The resulting form of xdxcsc5 fromthe rule above requires the nexttechnique.


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2.3 Trigonometric Substitution (TC7 pp. 594-599 / TCWAG pp. 545-550)

For integrals containing, 22 xa , 22 ax + and 22 ax , 0>a .

Or, 22 ua , 22 au + and 22 au where u is a differentiable function of .

MUST REMEMBER!!! For xdxsectan nm and xdxcscxcot nm .CASE 1. n is even. Separate xsec2 or xcsc2 . Express the rest in terms an or

cot using xtanxsec 22 1+= or xcotxcsc 22 1+= . Proceed withsubstitution!

CASE 2. m is odd. Separate one factor tan or cot and one factor of sec or cscc .Express the rest in terms of sec or xcsc .Proceed with substitution!


1. xdxsecxtan 42 2. xdxcscxcot 33

EXERCISE.


1. xdxsecxtan 32 2. dxcscxcot 43 3. xdxsecxtan 44

REMEMBER!!! When all efforts and all else fail,

when odd-powered, separate one factor when even-powered, separate two factors what you separate is a derivative of some other function


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For 22 ua , let For 22 au + , let

For 22 au , let

REMEMBER!!!

for 22 xa for 22 ax + for 22 ax

MUST REMEMBER!!! Substitutes to use . . .

for 22 xa , let

for 22 ax + , let

for 22 ax , let


1. dx

x

x29

2. dx

x

x 162


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2.4 Integration Using Partial Fractions (TC7 pp. 601-603 / TCWAG pp. 551-565)

For intergrals of the form( )( ) dxxQxP

where and Q are polynomials such that QdegPdeg < .

If QdegPdeg , simplify firstQ

Peither by factoring or division

of polynomials.

GivenQ

Pin its simplest form,

Q

Pcan be decomposed to a sum of partial fractions.

n

n

Q

P...

Q

P

Q

P

Q

P

Q

P++++=

3

3

2

2

1

1 where { }nQ,...,Q,Q,QLCDQ 321=

and ii QdegPdeg < for each i .

RESTRICTIONS (for discussion)!

Consider only linear and quadratic factors of Q . Hence, partial fractions will be of the form:

bax

A

+ or

cbxax

BAx

++

+

2

EXERCISE.


1. dxxx 23 25 2. + 249 x

dx 3.

2522 xx

dx

MUST REMEMBER!!!

CASE 1. The factors of Q are distinct.

If the factors of Q are all linear,nQ

D...

Q

C

Q

B

Q

A

Q

P++++=

321.

If the factors of Q are all quadratic,

n

nn

Q

BxA...

Q

BxA

Q

BxA

Q

BxA

Q

P +++

++

++

+=

3

33

2

22

1

11 .

3.

( ) + 2312xdx


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TO DO!!! Evaluate the following by decomposing the integrand to a sum of partial fractions.

1. +dx

xx

x

6

17

2

Part A. Part B.

2. ++dxxxx

x

2

23

23

Part A. Part B.

3.( )( ) +

+dx

xx

xx

11

123

2

2

Part A. Part B.

EXERCISE.


1. +dx

x

xx

3

2 14 2.

( )( ) +++dx

xx

x

31

17

2 3.

( ) ( ) +++

dxxxx

x

222

6

2

2

___________________After decomposing to partial fractions, you need to express #2 as a sum of three integrals. For #3,

you need to do some completing of squares to solve the form ++ 222 xxdx

.


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MUST REMEMBER!!!

CASE 2. Q has some repeating factors.

If a is a repeating factor of Q such that ( )max is a factor, the partial fraction

should contain( ) ( ) ( )max

D...axC

axB

axA

++

+

+

32.

If cbxax ++2 is a repeating factor of Q such that ( )mcbxax ++2 is a factor, the partialfraction should contain

( ) ( ) ( )mmm

cbxax

BxA...

cbxax

BxA

cbxax

BxA

cbxax

BxA

++

+++

++

++

++

++

++

+

232

33

22

22

2

11 .

TO DO!!!

Evaluate the following by decomposing the integrand to a sum of partial fractions.

1. ( ) ( ) ++ 212 xx

dx

Part A. Part B.

2.

( ) +dx

xx

x

22

2

1

2

Part A. Part B.

EXERCISE.


1.( ) dx

xx

x

22 1

2 2.

( ) + 32 1xdx

3. +++

dxxxx

x

234 44

3


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(Supplement) Other Substitution Techniques (TC7 pp. 614-619 / TCWAG pp. 566-569)

For rational functions of sine and cosine, use the substitute:

=2

1tanz =cos , =sin , =d

For varying rational exponents or radicals, use the substitute:

nzx= where n is the proper exponent to remove all rational exponents inthe resulting integral

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2.5 Improper Integrals (TC7 pp. 650-664 / TCWAG pp. 665-676)

REVIEW!!! Evaluating definite integrals.

If f is continuous on [ ]b,a and ( ) ( ) CxFdxxf += , then ( ) ( ) ( )aFbFdxxfb

a= .

TO DO!!!

Evaluate the following by decomposing the integrand to a sum of partial fractions.

1. +

2cossin

d 2.

tansin

d

TO DO!!!

Evaluate the following using proper substitutes.

1. + dxxx3

1 Use 3zx= .

2. + dxxx 425 Use 422 +=xz .

MUST REMEMBER!!!

For integrals over unbounded intervals: ( b, , )+,a or ( )+ ,

Assume that f is continuous within the interval of integration,

( ) ( ) +

+

=t

atadxxflimdxxf ( ) ( )

=

b

tt

b

dxxflimdxxf

( ) ( ) ( ) +

+

+=

0

0

dxxfdxxfdxxf

If the respective limit(s) exists and is finite, the improper integral is said to be convergent.Else, it is divergent.


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TO DO!!!

Determine whether the given improper integral is convergent or divergent.

1.( )

+

0321 x

dx

2. +0

31x

dx

3. +

+21 x

dx

MUST REMEMBER!!!

For integrals of functions with infinite discontinuity over bounded intervals:

Assume that f is continuous on ( b,a and ( ) =+xflim

ax, ( ) ( ) +=

b

tat

b

adxxflimdxxf

Assume that f is continuous on [ )b,a and ( ) =xflim

bx, ( ) ( ) =

t

abt

b

adxxflimdxxf .

Assume that f is continuous on [ ]b,a except at ( )b,ac , ( ) += xflimcx ,

( ) ( ) ( ) +=b

c

c

a

b

adxxfdxxfdxxf .

If the respective limit(s) exists and is finite, the improper integral is said to be convergent.Else, it is divergent.


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END OF UNIT 2 Lecture Guide

TO DO!!!

Determine whether the given improper integral is convergent or divergent.

1.

2

0 24 x

dx

2. +0

2 2x

dx

3.

( ) 3

0 32

1x

dx

EXERCISE.

Establish convergence/divergence.

Case: Infinite discontinuity over interval of integration.

1. dx

x

x

3

02

9

2.

( ) +

0

12

1x

dx 3.

1

8 31

x

dx 4.

1

1

dx

x

e x

EXERCISE.

Establish convergence/divergence.

Case: Unbounded interval of integration.

1. 0

dxxex 2. +

1

dxx

xln 3.

+

dxxe x2

4. +

+dxx

x

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MATH 27 Lecture Guide UNIT 2