Indefinite Integration - Product-To-Sum - Questions

8/16/2019 Indefinite Integration - Product-To-Sum - Questions

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The Nail It Series Indefinite Integration

Product-to-Sum Formulae

Questions Compiled by:

Dr Lee Chu KeongNanyang Technological University

http://ascklee.org/CV/CV.pdfhttp://ascklee.org/CV/CV.pdf


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About the Nail It Series

About the Nail It Series

Nail It is a series of ebooks containing questions on various topics in mathematics, compiled

from textbooks that are out-of-print. Each ebook contains a collection of questions sufficient toensure that mastery is achieved for that topic. The idea behind the series is threefold:

(i) First, to give students sufficient practice on solving questions that are commonly asked

in examinations. Mathematics is not a spectator sport, and students need all the drill

they can get to achieve mastery. Nail It ebooks supplies the questions.

(ii) Second, to expose students to a wide variety of questions so that they can spot patterns

in their solution process. Students need to be acquainted with the different ways inwhich a question can be posed.

(iii) Third, to build the confidence of students by arranging the questions such that the easy

ones come first followed by the difficult ones. Confidence comes with success in solving

problems. Confidence is important because it leads to a willingness to attempt more

questions.

Finally, to “nail” something is to get it absolutely right, i.e., to master it. Nail It ebooks to enable

motivated students to master the topics they have problems with.

If you have any comments or feedback, I’d like to hear them. Please email them to me at

[email protected]. Finally, I’d like to wish you all the best for your learning journey.

Lee Chu Keong (May 14, 2016)

mailto:[email protected]:[email protected]:[email protected]


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Features of the Nail It Ebooks

Features of the Nail It Series Ebooks

1. The Nail It Series ebooks are completely free. The questions are compiled from textbooks

that are out-of-print and those that are hard to locate. As Winston Churchill once said,“We make a living by what we get. We make a life by what we give.”

2. The Nail It Series ebooks have been designed with mastery of the subject matter in mind.

There are plenty of textbooks, and they all can help you get the “A” grade. Nail It ebooks

are designed to make you the Michael Phelps of specific topics.

3. Each Nail It Series ebook has a collection of questions which are deemed necessary to

master a topic. Each question appears on its own page. View it on your tablet or a mobilephone, and start working on them.

4. The Nail It Series ebooks are modular, and compatible with different syllabi used in

different parts of the world. I list down the links with the syllabi I am familiar with.

5. Students are usually engrossed in solving questions, and miss out on the connections

between different questions. Compare pages puts the spotlight on usually two, but

sometimes more questions, the solution of which are closely related. Contrast pages doesthe same, but with two or more questions that look alike, but that require different

approaches in its solution. Spot the Pattern pages challenge students to spot the pattern

underlying the solution process.

6. Essential to Know pages provides must-know facts about questions already completed. I

suggest committing the material presented in the Essential to Know pages to memory.


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About Learning

words, are you happy with 75 marks, or would not be satisfied until you get 100 marks? A

gulf separates an “A” grade from complete mastery, and you have to decide what you are

after. This is because the game plan for each is different.

6. Learning takes participation. There are no “passengers” in learning. It is immersive, and

requires you to be interested, alert, and engaged.

7. Learning takes courage. It requires you ask people for help, step out of your comfort zone,

re-examine your assumptions, and make mistakes. All this takes courage, and requires

you to step out of your comfort zone. Are you courageous enough to learn?

This begs the question: Did my teachers lie? Yes and no. What they were probably referring to

(as being fun and enjoyable) is the ecstasy one feels when mastery of a topic has been achieved.When you work hard for something, and you succeed, the feeling is simply indescribable. This

is why I encourage you to strive for mastery – it’s fun when you can solve everything on a topic

that’s thrown at you. The journey, however, is arduous and treacherous. Be prepared to slog.


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Syllabi Compatibility

Syllabi Compatibility

The contents of this Nail It ebook will benefit:

junior college students in Singapore, who are sitting for the GCE A Level H2 Mathematics

(9740) Paper;

Sixth Form students in Malaysia, who are sitting for the STPM Mathematics T (954) Paper;

students in India who are sitting for the IIT JEE (Main & Advanced) Mathematics Paper;

students around the world, who are sitting for the Cambridge International Examinations

(CIE) Mathematics Paper.


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Indefinite Integration Using Product-to-Sum Formulae

Questions compiled by Dr Lee Chu Keong

Product-to-Sum Formulae

sin cos = 12 [sin + + sin − ]

cos sin = 12 [sin + − sin − ]

cos cos = 12 [cos + +cos − ]

sin sin = 12 [cos + −cos − ]


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Section I

Sine

Sine


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Question 1

Find:

∫sin 52 sin32 d

Source: LC262(7)


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Question 2

Find:

∫sin3sind Source: PV380(28)


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Question 3

Find:

∫sin3sin2d Source: AM296(10), JG57(2i)


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Question 4

Find:

∫sin4sin2d


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Question 5

Find:

∫sin5sind Source: AM302(42)


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Question 6

Find:

∫sin5sin2d


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Question 7

Find:

∫sin5sin4d


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Question 8

Find:

∫sin6sin4d Source: JLS480(18)


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Question 9

Find:

∫sin7sin3d Source: AM292(9)


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Question 10

Find:

∫sin8sin3d


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Section II

SineCosine


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Question 11

Find:

∫sincos2 d


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Question 12

Find:

∫sin(32 ) cos 2 d

Source: PV401(25)


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Question 13

Find:

∫sin2cosd Source: LC231(7)


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Question 14

Find:

∫sin3cosd


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Question 15

Find:

∫sin4cos3d Source: RLR61(v)


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Question 16

Find:

∫sin5cos3d


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Question 17

Find:

∫sin6cos2d Source: JLS479


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Question 18

Find:

∫sin7cos3d Source: AM292(8)


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Section III

CosineSine


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Question 19

Find:

∫cos3sin4 d


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Question 20

Find:

∫cos2sind Source: AET355(15a)


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Question 21

Find:

∫cos(52 ) sin 2 d


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Question 22

Find:

∫cos3sin2d Source: AET353(4), JG57(2ii)


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Question 23

Find:

∫cos4sin2d Source: AM302(40)


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Question 24

Find:

∫cos4sin3d Source: AET355(15b)


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Question 25

Find:

∫cos5sin2d Source: DDB425(10)


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Question 26

Find:

∫cos5sin3d Source: AM296(11)


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Question 27

Find:

∫cos5sin4d Source: PV380(21)

d f d l


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Question 28

Find:

∫cos7sin2d Source: DDB426(30)

I d fi it I t ti U i P d t t S F l


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Section IV

CosineCosine

I d fi it I t ti U i P d t t S F l


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Question 29

Find:

∫coscos2 d

Indefinite Integration Using Product to Sum Formulae


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Question 30

Find:

∫cos2cosd Source: RLR63(xv)



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Question 31

Find:

∫cos3cos2d Source: AM302(41)



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Question 32

Find:

∫cos4cosd Source: PV380(22)



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Question 33

Find:




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Question 34

Find:

∫cos5cos3d Source: DDB426(29), JLS481(19)



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g g


Question 35

Find:




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Question 36

Find:

∫cos7cos5d



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Question 37

Find:

∫cos15cos4d



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Section V

Miscellaneous



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Question 38

Find:

∫sinsin2sin3d



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Question 39

Find:

∫sinsin3sin5d



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Question 40

Find:

∫coscos2sin4d



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Question 41

Find:

∫sin2cos5sec4 d



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Question 42

Find:

∫cos sin8d


i


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Question 43

Find:

∫sin cos4d Source: AET355(15c)


Q i 44


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Question 44

Find:

∫cos 2sin3 Source: AET355(15d)

Sources

Sources


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Sources

AY Ayres, F., & Mendelson, E. (2000). Calculus (4th ed.). New York: McGraw-Hill.

DDB Berkey, D.D. (1988). Calculus (2nd ed.). New York: Saunders College Publishing.

EP Edwards, C.H., & Penney, D.E. (1986). Calculus and Analytic Geometry (2nd ed.).

Englewood Cliffs, NJ: Prentice-Hall.

EWS Swokowski, E.W. (1984). Calculus with Analytic Geometry (3rd ed.). Boston, MA: Prindle,

Weber & Schmidt.

JG Gilbert, J. (1991). Guide to Mathematical Methods. Basingstoke, England: Macmillan.

JLS Smyrl, J.L. (1978). An Introduction to University Mathematics. London: Hodder and

Stoughton.

JMAW Marsden, J., & Weinstein, A. (1985). Calculus I . New York: Springer-Verlag.

GM Matthews, G. (1980). Calculus (2nd ed.). London: John Murray.

GS Strang, G. (1991). Calculus. Wellesley, MA: Wellesley-Cambridge Press.

LS Chee, L. (2007). A Complete H2 Maths Guide (Pure Mathematics). Singapore: Educational

Publishing House.MW March, H.W., & Wolff, H.C. (1917). Calculus. New York: McGraw-Hill Co.

PV Purcell, E.J., & Varberg, D. (1987). Calculus with Analytic Geometry (5th ed.). Englewood

Cliffs, NJ: Prentice-Hall.

Sources

RAA Adams, R.A. (1999). Calculus: A Complete Course (4th ed.). Don Mills, Canada: Addison


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RAA Adams, R.A. (1999). Calculus: A Complete Course (4th ed.). Don Mills, Canada: Addison

Wesley Longman.

RCS Solomon, R.C. (1988). Advanced Level Mathematics. London: DP Publications.

RIP Porter, R.I. (1979). Further Elementary Analysis (4th ed.). London: G. Bell & Sons.RLR Rosenberg, R.L. (1984). Elementary Calculus: Course Notes. Ottawa, Canada: Holt,

Rinehart and Winston.

SIG Grossman, S.I. (1988). Calculus (4th ed.). Harcourt Brace Jovanovich.

SRG Sherlock, A.J., Roebuck, E.M., & Godfrey, M.G. (1982). Calculus: Pure and Applied.

London: Edward Arnold.

TFWG Thomas, G.B., Finney, R.L., Weir, M.D., & Giordano, F.R. (2003). Thomas’ Calculus (Updated 10th ed.). Boston: Addison Wesley.

TKS Teh, K.S. (1983). Pure and Applied Mathematics (‘O’ Level). Singapore: Book Emporium.

TWS Tan, W.S. (1976). Ilmu Hisab Tambahan (Jati) (2nd ed.). Kuala Lumpur, Malaysia:

Dewan Bahasa dan Pustaka.

WFO Osgood, W.F. (1938). Introduction to the Calculus.

About Dr Lee Chu Keong



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Dr Lee has been teaching for the past 25 years. He has taught in

the Nanyang Technological University, Temasek Polytechnic, and

Singapore Polytechnic. The excellent feedback he obtained year afteryear is a testament to his effective teaching methods, the clarity with

which he explains difficult concepts, and his genuine concern for the

students. In 2015, Dr Lee won the Nanyang Teaching Award (School

Level) for dedication to his profession.

Dr Lee has a strange hobby – he collects mathematics textbooks.

He visits bookstores when he goes to a city he has never been to, to

find textbooks he does not already have. So far, he has textbooksfrom Singapore, China, Taiwan, Japan, England, the United States,

Malaysia, Indonesia, Thailand, Myanmar, France, the Czech Republic,

France, India and Australia. The number of textbooks in his

collection grows practically every week!

For mathematics, Dr Lee believes the only way to better grades is practice, more practice,

and yet more practice. While excellent textbooks are a plenty, compilations of questions are alot harder to find. For this reason, he started the Nail It Series, a series of ebooks containing

questions on various topics commonly tested in mathematic examinations around the world.

Carefully studying the questions and working their solutions out should improve the grades of

the students tremendously.

Documents

Indefinite Integration - Product-To-Sum - Questions