teach yourself maths - student xpress

TEACH YOURSELF MATHS

LEAVING CERTIFICATE HONOURS

77 LESSONS

WEB SOLUTIONS FOR PAPERS 2001 TO 2006EVERY FORMULA INCLUDED

EASY TO FOLLOW STEPS

FULL LIST OF PROOFS

STUDY PLAN PROVIDED

Tony KellyKieran Mills

EVERYTHING YOU NEED TO KNOW

THE EASY WAY TO LEARN MATHS

MATHS IS ABOUT DOING QUESTIONS AND GETTING THE

RIGHT ANSWER. YOU LEARN MATHS BY DOING. IT IS

LIKE PLAYING A SPORT OR A MUSICAL INSTRUMENT –THE MORE YOU PRACTISE THE BETTER YOU GET AND

THE MORE ENJOYMENT YOU GET .

© Tony Kelly & Kieran Mills

Printed by Print4less

Published by Student Xpress 2006www.studentxpress.ie

All rights reserved.No part of this publication may be reproduced, copiedor transmitted in any form or by any means withoutwritten permission of the publishers or else under theterms of any licence permitting limited copying issuedby the Irish Copyright Licensing Agency, The Writers’Centre, Parnell Square, Dublin 1.

HOW DO YOU USE THIS BOOK?

1. HAVE A LOOK AT THE LAY-OUT OF THE LEAVING CERT. PAPER

PAPER 1 (Do 6 out of 8)1. Algebra2. Algebra3. Complex Numbers & Matrices4. Sequences & Series5. Sequences & Series6. Differentiation & Applications7. Differentiation & Applications8. Integration

PAPER 2SECTION A (Do 5 out of 8)1. Circle2. Vectors3. Line4. Trigonometry5. Trigonometry6. Discrete Maths7. Discrete MathsSECTION B (Do 1 out of 4)8. Further Calculus

You can start studying any section you wish but it is advisable to start withAlgebra and Trigonometry as these areas contain the fundamentals that areused in most other sections.

2. STUDY PLAN

Once you have selected your question go to the study plan so you can keeptrack of your learning by filling in the boxes as you progress.

3. STARTING THE LESSON

Now start reading through the lesson. Have pen and paper at the ready writingdown all the formulae. Many of the formulae you need to use are contained inthe official Department of Education table book [Pages 6, 7, 9, 41 and 42]. Theformulae you need to know for the Leaving Cert. from the table book pagesare at the end of this book. There is a colour code for formulae:

...... 8

...... 8

...... 8

Formula in the table book

Similar formula in the table book

Formula you need to learn

COMPLEX NUMBERS & MATRICES (Question 3, Paper 1)

1. Complex Number Algebra2. Complex Number Equations

LESSONS START PROOFS/FORMULAE LC QUESTIONS FINISH

3. De Moivre’s Theorem4. Matrix Algebra5. Matrix Equations

10/12 (4.00) 10/12 (6.00)

11/12 (4.00) 11/12 (6.00)

12/12 (5.00) 10/12 (8.00)

Now see if you can do the Leaving Cert. questions. All the parts of the Leav-ing Cert. from 2001 to 2006 based on each lesson are listed at the end of thelesson. You can get them on the website along with their solutions.Do the questions on your own and then check the solutions. If you get a wronganswer or get stuck then start again. You have only succeeded when you cancomplete each question from start to finish on your own.Be honest with yourself. You know when you have succeeded. Feel the satis-faction that you are accomplishing your tasks. It is hard work and takes a longtime. But, at least you have a plan and you know how to go forward to suc-ceed. Tick the study plan when you are finished.

4. START DOING QUESTIONS

WEBSITE: www.studentxpress.ieThis website is a valuable resource containing lots ofmathematical material. All the Leaving Cert. papers from2001 to 2006 are on the website with all the solutions. Thebook can also be purchased online.

Use this book anytime you are doing Maths. It contains everything you need toknow to do the Leaving Cert. honours papers. So get working and enjoy.

PROOFThe proofs needed for the Leaving Cert. are inboxes as shown. Learn any proofs in the lessonand learn formulae with a red circle. Tick thestudy plan when you know them.

If you need more practice and detail you can get extra help from the two textbooks on which this revision book is based. These books are available inEason bookshops and online (www.studentxpress.ie).The Magical Bag of Mathematical Tricks, Paper 1The Magical Bag of Mathematical Tricks, Paper 2

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1

ALGEBRA

ALGEBRA (Q 1 & 2, PAPER 1)

LESSON No. 1

IN THIS LESSON YOU WILL REVISE THE BASICS OF ALGEBRA.

SOME BASICS

1. Difference of 2 squares[B] FACTORS: You need to remember these factors.

[A] MULTIPLYING OUT BRACKETS

....... 1a b a b a b2 2− = + −( )( )

Ex. x y x y x y x y2 2 2 281 9 9 9− = − = + −( ) ( ) ( )( )

3. Difference of 2 cubes

2. Sum of 2 cubes

a b a b a ab b3 3 2 2+ = + − +( )( ) ....... 2

Ex. x y x y x y x xy y3 3 3 3 2 28 2 2 2 4+ = + = + − +( ) ( ) ( )( )

a b a b a ab b3 3 2 2− = − + +( )( ) ....... 3

Ex. 8 27 2 3 2 3 4 6 93 3 3 3 2 2x y x y x y x xy y− = − = − + +( ) ( ) ( )( )

[C] MULTIPLICATION OF FRACTIONS

Ex. ( )( )

( )( ) ( )( )

a ba b

b ab a

a ba b

b ab a b a

++

×−−

=++

×−

+ −=

2

2 2

2

1

Multiply tops and multiply bottoms and/or cancel. Remember to factoriseeverything.

It is very useful to be able to expand squares and cubes of brackets.

( ) ( )( )x y x y x y x xy y+ = + + = + +2 2 22 [Coefficients: 1, 2, 1]

( )x y x x y xy y+ = + + +3 3 2 2 33 3 [Coefficients: 1, 3, 3, 1]

2

ALGEBRA

[E] POWERS

1. a a am n m n× = +

2. aa

am

nm n= −

3. a0 1=

POWER RULES

7. ab

ab

m

n

p mp

np

⎛

⎝⎜

⎞

⎠⎟ =

8. ab

ba

m

n

p n

m

p⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟

−

[F] SURDS

Ex. 23 1

23 1

3 13 1

2 3 12

3 1−

=−

×++

=+

= +( )

www.studentxpress.ie/papers.htm

LEAVING CERT. QUESTIONS

LESSON 1: SOME BASICS

2006 1 (a)2005 1 (b)2004 1 (a), 1 (b) (ii), 1 (c)2003 1 (a)2001 1 (a), 2 (b) (ii)

Ex. 41

72 3 5

41 1

72 5 12 2x x x x x x x−

−− −

=+ −

−− +( )( ) ( )( )

[D] ADDING FRACTIONS

You must use the lowest common denominator. Remember to factorise thedenominators first.

=− − −

− + −=

−− + −

4 2 5 7 12 5 1 1

132 5 1 1

( ) ( )( )( )( ) ( )( )( )

x xx x x

xx x x

4. aa

nn

− =1

5. ( )a am n mn=

6. ( )a b a bm n p mp np=

Ex. Express 3 3 315

15

15+ + in the form 3

pq where p q, .∈Z

SOLUTION

3 3 3 3 3 315

15

15

15

65+ + = × =

You are often asked to rationalise the denominator of a surd. This meansgetting rid of surds in the denominator. Multiply above and below by theconjugate of the denominator.

3

ALGEBRA

Ex. If x bx c2 + + is a factor of x p3 − show that (i) b p3 = , (ii) bc = p.SOLUTION

The long division process is useful in the type of question outlined in the example.

The remainder has to be zero ⇒ = +R x0 0

⇒ = =b c bc p2 and . Now prove what they are asking you.

If ( )x k− is a factor of f (x) then k is a root of f (x) = 0,i.e. f (k) = 0 and vice versa.

The factor theorem states that:

x b−x x x p3 20 0+ + −

∓ ∓ ∓x bx cx3 2

− − −bx cx p2

± ± ±bx b x bc2 2

( ) ( )b c x bc p2 − + −

LESSON No. 2

IN THIS LESSON YOU WILL LEARN THE DIVISION PROCESS WHICH IS USEFUL IN OBTAINING

CERTAIN RESULTS.

DIVISION


LESSON 2: DIVISION

2006 1 (c) 2005 1 (c)2003 2 (b) (ii) 2001 1 (c)

LESSON No. 3

IN THIS LESSON YOU WILL LEARN ABOUT THE FACTOR THEOREM. THIS THEOREM PROVIDES AMETHOD FOR SOLVING ALL KINDS OF EQUATIONS.

FACTOR THEOREM

x bx c2 + +

Ex. ( )x −1 is a factor of f x x x x( ) .= + − −2 2 13 2

∴ =x 1 is a root or a solution of f x( ) = 0

∴ = + − − = + − − =f ( ) ( ) ( ) ( )1 2 1 1 2 1 1 2 1 2 1 03 2


This type of problem can also be done by lining up coefficients. Both methodsare outlined on the website.

4

ALGEBRA

PROOF OF FACTOR THEOREM

f x ax bx cx d( ) = + + +3 2

f k ak bk ck d( ) = + + +3 2

∴ − = − + − + −f x f k a x k b x k c x k( ) ( ) ( ) ( ) ( )3 3 2 2

= − + + + + +( ){ }x k ax akx ak bx bk c2 2 = −( ) ( )x k g x

∴ = + −f x f k x k g x( ) ( ) ( ) ( )

(i) f k f x x k g x( ) ( ) ( ) ( )= ⇒ = −0 ∴ −x k is a factor.

(ii) x k− is a factor ⇒ =f k( ) .0

Can you prove the factor theorem for a quadratic equation f x ax bx c( ) ?= + +2

[B] PROPERTIES OF ROOTS α β,

These are equations of the form y f x ax bx c= = + + =( ) .2 0[A] METHODS OF SOLUTION

x b b aca

=− ± −2 4

2

You need to be able to prove the factor theorem.


LESSON 3: FACTOR THEOREM

2006 1 (b) 2004 1 (b) (i)2003 1 (b) 2001 1 (b)

LESSON No. 4

IN THIS LESSON YOU WILL LEARN ABOUT QUADRATIC EQUATIONS, HOW TO MAKE NEW

EQUATIONS USING THE ROOTS AND HOW TO INTERPRET THE MEANING OF THE ROOTS.

QUADRATIC EQUATIONS

Sum S: α β+ = − =−b

a21

nd.

st.

Product P: αβ = =ca

31

rd.

st.

You can solve quadratic equations by1. Factorisation: Works sometimes, OR

2. Use formula 4: Always works. ....... 4

....... 5

....... 6


5

ALGEBRA

The following results are useful whengenerating new quadratics from oldquadratics:

Forming a quadratic equation given its roots:

x x2 0− + =S P

Ex. Form a quadratic equation that has roots of 3 and −1.SOLUTION

Roots: α β= = −3 1,

Sum S: α β+ = − =3 1 2 and Product P: αβ = − = −( )( )3 1 3

Equation: x x x x2 20 2 3 0− + = ⇒ − − =S P

α β α β αβ2 2 2 2+ = + −( )

α β α β α αβ β3 3 2 2+ = + − +( )( )

α β α β α β4 4 2 2 2 2 22+ = + −( )

[C] NATURE OF THE ROOTS AND GRAPHS OF QUADRATICS

1. If b ac2 4 0− > ⇒ ≠α β[Two different real roots.]

2. b ac2 4 0− = ⇒ =α β[Two real roots that are the same.]

3. b ac2 4 0− < ⇒α β, not real.[There are no real roots (complex roots).]

X

Y

X

Y

X

Y

α β=

βα

....... 7

REMEMBER: If b ac2 4 0− ≥ ⇒ Real roots.

If b ac2 4 0− < ⇒ Unreal or complex roots.

The nature of the roots is determined by the expression under the square rootsign in formula 4. It is b ac2 4− and is called the DISCRIMINANT.

6

ALGEBRA


LESSON 5: CUBIC EQUATIONS

2005 2 (b) 2002 1 (b)

1. Guess at a root α (it must divide exactly into the constant term).2. Form a factor from the root ( ).x −α3. Factorise fully using division process and solve.

STEPS


LESSON 4: QUADRATIC EQUATIONS

2006 2 (b)2004 2 (b) (ii), 2 (c) (ii)2003 1 (c), 2 (c) (ii)2002 1 (a), 1 (c), 2 (c)2001 2 (c)

These are equations of the form y f x ax bx cx d= = + + + =( ) .3 2 0Method of solution: Guess at a root using the factor theorem.The basic method for solving a cubic equation is:

LESSON No. 5

IN THIS LESSON YOU WILL LEARN HOW TO SOLVE CUBIC EQUATIONS. THIS INVOLVES GUESS-ING A ROOT AND THEN USING THE FACTOR THEOREM.

CUBIC EQUATIONS

Ex. Solve 2 2 1 03 2x x x+ − − = given it has an integer root.SOLUTION

1. f ( ) ( ) ( ) ( )1 2 1 1 2 1 1 2 1 2 1 0 13 2= + − − = + − − = ⇒ is a root.

2. ( )x −1 is a factor.

3. 2 2 1 1 2 3 13 2 2x x x x x x+ − − ÷ − = + +( ) by division.Solve the quadratic by factorisation or using the formula.

∴ + − − = − + + =2 2 1 1 1 2 1 03 2x x x x x x( )( )( )

∴ = − −x 1 112, ,



7

ALGEBRA


LESSON 6: FUNCTIONS

2006 2 (c) 2005 2 (c)2004 2 (c) (i) 2002 2 (b) (ii)

A function simply takes in a number (object), performs an operation on it andsends out a new number (object). The operation is determined by the rule of thefunction.

x Operation f x y( ) =

Function, f

LESSON No. 6

IN THIS LESSON YOU WILL LEARN ABOUT FUNCTIONS. ON THE LEAVING CERT. PAPER

FUNCTIONS ARE USED EXTENSIVELY AND APPLIED IN ALL KINDS OF SITUATIONS.

FUNCTIONS

Ex. Let f x x kmx

( ) ,=+2 2

where k and m are constants and m ≠ 0. Show

that f km f km

( ) .= ⎛⎝⎜

⎞⎠⎟

SOLUTION

f km km km km

k mkm

k mm

( ) ( )( )

( ) ( )=

+=

+=

+2 2 2 2

2

2

2

1 1

f km

km

kk

mm

k mm

km

km

m⎛⎝⎜

⎞⎠⎟ =

+=

+× =

+( )( )

( ) ( )2 2 2 1 2

2

2

2

2 1 1


8

ALGEBRA

[A] QUADRATICS: ax bx c2 0+ + ≤

STEPS

1. Get all terms on one side and zero on the other side.2. Solve the corresponding equation to get the roots α β, .3. Carry out the region test. Use the roots in ascending order to form

regions: ← ↔ →α β Choose a nice number in each region to testthe inequality using the test box.

4. Based on the region test write down the solutions.

LESSON No. 7

IN THIS LESSON YOU WILL LEARN ABOUT DEALING WITH THREE TYPES OF INEQUALITIES:QUADRATIC, RATIONAL AND MODULUS INEQUALITIES.

INEQUALITIES

Ex. Solve x x2 5 6> − .SOLUTION

1. x x2 5 6 0− + >

2. Solve x x x x x2 5 6 0 3 2 0 3 2− + = ⇒ − − = ⇒ =( )( ) ,

Roots: α β= =2 3,3. Region test:

0( )( )− − >3 2 0

Correct

2.5( )( )− ⋅ ⋅ >0 5 0 5 0

Wrong

4( )( )1 2 0>Correct

2 3

4. ∴ < >x x2 3,

Region Test on ( )( )x x− − >3 2 0 .....Test Box

[B] RATIONALS: P xQ x

( )( )

> 0

STEPS

1. Multiply both sides by the denominator squared unless you arecertain that it is positive.

2. Get all terms on one side and take out the highest common factor.3. Solve the corresponding equation.4. Do region test on the roots in ascending order on Test Box.5. Based on the region test write down the solutions.

9

ALGEBRA

Ex. Solve xx

x−+

< − ∈12

1, R.

SOLUTION

1. xx−+

< −12

1 ⇒ − + < − +( )( ) ( )x x x1 2 1 2 2 [Multipy by ( )x + 2 2 ]

2. ⇒ + − + + <( )[( ) ( )]x x x2 1 2 0

−3( )( )− − <1 5 0

Wrong

−1( )( )1 1 0− <

Correct

0( )( )2 1 0<

Wrong

3. Solve ( )( ) ,x x x+ + = ⇒ = − −2 2 1 0 2 12

Roots: α β= − = −2 12,

4. Region test:Region Test on ( )( )x x+ + <2 2 1 0 .....Test Box

−2 − 12

5. ∴− < < −2 12x

[C] MODULUS ax b c+ >

STEPS

1. Solve the corresponding modulus equality.2. Do region test on roots in ascending order on Test Box.3. Based on the region test write down the solutions.

Ex. Solve 2 7 4 1x x− ≤ − .SOLUTION

1. Solve 2 7 4 1 3 43x x x− = − ⇒ = − ,

2. Region test:

Region Test on 2 7 4 1x x− ≤ − .....Test Box

−4− ≤ −15 17

Correct

0

− ≤ −7 1Wrong

2

− ≤3 7Correct

−3 43

3. ∴ ≤ − ≥x x3 43,

10

ALGEBRA

STEPS

1. Eliminate one letter.2. Solve for the other.3. Substitute into either of the original equations to get

second unknown.

[A] 2 LINEARS (x, y)


LESSON 7: INEQUALITIES

2002 2 (b) (i) [Quadratic]2004 2 (b) (i) [Rational]2005 2 (a) [Modulus]2003 2 (b) (i) [Modulus]2001 2 (b) (i) [Modulus]

LESSON No. 8

IN THIS LESSON YOU WILL LEARN HOW TO SOLVE DIFFERENT TYPES OF SIMULTANEOUS

EQUATIONS.

SIMULTANEOUS EQUATIONS

2 3 5 2x y+ = − ×( )

3 2 12 3x y− = ×( )13 26 2x x= ⇒ =

9 6 36x y− =4 6 10x y+ = − 2 2 3 5( ) + = −y

⇒ = −y 3→ →

STEPS

1. Eliminate a letter from the linear.2. Substitute into quadratic and solve for the other letter.3. Substitute these values into linear to get all solutions.

[B] LINEAR AND QUADRATIC

→ →y x= −2 5

x xy2 2+ =⇒ = −x 1

3 2,⇒ − − =3 5 2 02x x

x x x2 2 5 2+ − =( ) y = − − = −2 513

173( )

y = − = −2 2 5 1( )

SOLUTIONS: ( , ), ( , )− − −13

173 2 1


11

ALGEBRA


LESSON 8: SIMULTANEOUS EQUATIONS

2005 1 (a) [2 linears]2004 2 (a) [3 linears]2002 2 (a) [3 linears]2006 2 (a) [Linear and Quadratic]2003 2 (a) [Linear and Quadratic]2001 2 (a) [Linear and Quadratic]

[C] 3 LINEARS (x, y, z)

STEPS

1. Eliminate one letter using two equations.2. Eliminate the same letter using two others.3. Solve the resulting equations as for two equations in two unknowns.4. Work backwards to find all letters.

→

x y zx y zx y z

− + = −+ − =− − =

3 4 1 12 3 8 23 2 5

.......( ).........( )..........( )3

x y zx y z− + = −+ − = ×3 4 1 1

6 3 9 24 2 3.......( ).....( )

7 5 23 4x z− = .....( )

4 2 6 16 2 23 2 5 3

x y zx y z+ − = ×− − =

......( )..........( )

7 7 21 5x z− = .....( )

7 7 21 5x z− = .....( )

7 5 23 4x z− = .....( )SOLUTIONS: x y z= = =4 3 1, ,


12

ALGEBRA

QUICK OVERVIEW OF ALGEBRA

....... 1a b a b a b2 2− = + −( )( )

a b a b a ab b3 3 2 2+ = + − +( )( ) ....... 2

a b a b a ab b3 3 2 2− = − + +( )( ) ....... 3

If ( )x k− is a factor of f (x) then k is a root of f (x) = 0,i.e. f (k) = 0 and vice versa.

x b b aca

=− ± −2 4

2

Sum S: α β+ = − =−b

a21

nd.

st.

Product P: αβ = =ca

31

rd.

st.

α β α β αβ2 2 2 2+ = + −( )

α β α β α αβ β3 3 2 2+ = + − +( )( )

α β α β α β4 4 2 2 2 2 22+ = + −( )

....... 4

....... 5

....... 6

....... 7

REMEMBER: If b ac2 4 0− ≥ ⇒ Real roots.

If b ac2 4 0− < ⇒ Unreal or complex roots.

x x2 0− + =S P

Difference of two squares

Sum of two cubes

Difference of two cubes

Factor Theorem (Proof required)

Formula for solving quadratic equations

Roots of Quadratic Equations

Forming a Quadratic Equation

130

THE TABLES

THE TABLES

You are allowed to use the official Department of Education tablebook in the exam hall. There is a lot of information in this book thatyou do not need. The information on the following pages has beenextracted from the official table book and is exactly what you needfor the Leaving Cert. Honours Maths papers.

131

THE TABLES

PAGE 6 & 7 OF THE TABLES

r

C

b

a

ch

C

b

a

h

h

r

r

h

r

l

CIRCLE

Length = 2π rArea = π r 2

SECTOR

Length = rθ θ( in radians)

Area = 12

2r θ θ( in radians)

TRIANGLE

Area = ahArea = ab Csin

PARALLELOGRAM

CYLINDER

Area of curved surface = 2π rhVolume = π r h2

CONE

Curved surface area = π rl

Volume = 13

2π r h

SPHERE

Area of surface = 4 2π r

Volume = 43

3π r

Area = 12 ah

Area = 12 ab Csin

r θ

132

THE TABLES

cos( ) cos cos sin sinA B A B A B+ = −

sin( ) sin cos cos sinA B A B A B+ = +

tan( ) tan tantan tan

A B A BA B

+ =+

−1

Compound Angle formulae

The formulae for cos( ), sin( ),A B A B− −

tan( )A B− can be obtained by changing thesigns in these formulae. You need to be able toprove cos( )A B± and sin( ).A B±

cos sin2 2 1A A+ =

tan sincos

A AA

=

sec tancos

2 221 1A A

A= + =

cottan

AA

=1

seccos

AA

=1

cosecsin

AA

=1

There are 6 trig functions. They can all be written in terms of sine and cosine.

A 0 π2π π

3π4

π6

cos A

sin A

tan A 0

0

1 −1

0

0

0

1

∞

12

32

3

12

12

1

32

12

13

A 0o 180o 90o 60o 45o 30o

cos( ) cos− =A A sin( ) sin− = −A A tan( ) tan− = −A A

This is how you deal with negative angles.

Use the Sine and Cosine rules to solve triangles.

Sine formula: a

Ab

Bc

Csin sin sin= =

Cosine formula: a b c bc A2 2 2 2= + − cos

[Radians]

[Degrees]

(cos , sin )A A

cos A

sin A

1

1

-1

-1

A

[Prove]

A

CB

bc

a

PAGE 9 OF THE TABLES

133

THE TABLES

cos cos sin2 2 2A A A= −

sin sin cos2 2A A A=

tan tantan

2 21 2A A

A=

−

These formulae are obtainedby replacing B by A in thecompound angle formulae.

cos tantan

2 11

2

2A AA

=−+

sin tantan

2 21 2A A

A=

+

cos ( cos )2 12 1 2A A= + sin ( cos )2 1

2 1 2A A= −

Use these when integrating trig squares.

Used to change products to sums, useful when integrating products of trig functions.

Used to change sums into products, useful when solving trig equations.

De Moivre’s Therorem

(cos sin ) cos sinθ θ θ θ+ = +i n i nn

2cos cos cos( ) cos( )A B A B A B= + + −

2sin cos sin( ) sin( )A B A B A B= + + −

2sin sin cos( ) cos( )A B A B A B= − − +

2cos sin sin( ) sin( )A B A B A B= + − −

sin sin sin( ) cos( )A B A B A B+ = + −2 2 2

sin sin cos( )sin( )A B A B A B− = + −2 2 2

cos cos cos( )cos( )A B A B A B+ = + −2 2 2

cos cos sin( )sin( )A B A B A B− = − + −2 2 2

134

THE TABLES

DIFFERENTIATION INTEGRATION

f x( ) ′ ≡f x ddx

f x( ) [ ( )]

xn nxn−1

ln x1x

cos x −sin xsin x cos xtan x sec2 xsec x sec tanx xcosec x −cosec cotx xcot x −cosec2 x

ex ex

eax aeax

sin−1 xa

12 2a x−

tan−1 xa

aa x2 2+

Products and Quotients:

y uv dydx

u dvdx

v dudx

= = +;

y uv

dydx

v dudx

u dvdx

v= =

−; 2

We take a > 0 and omit constants ofintegration.

f x( ) f x dx( )∫

x nn ( )≠ −1xn

n+

+

1

1

1x ln x

cos x sin xsin x −cos x

tan x ln sec x

ex ex

eax1a

eax

12 2a x−

sin−1 xa

12 2a x+

1 1

axa

tan−

cos2 x 12

12 2[ sin ]x x+

sin2 x 12

12 2[ sin ]x x−

Integration by parts:

u dv uv v du∫ ∫= −

PAGE 41 & 42 OF THE TABLES

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