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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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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Section I
Sine
Sine
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 1
Find:
∫sin 52 sin32 d
Source: LC262(7)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 2
Find:
∫sin3sind Source: PV380(28)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 3
Find:
∫sin3sin2d Source: AM296(10), JG57(2i)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 4
Find:
∫sin4sin2d
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 5
Find:
∫sin5sind Source: AM302(42)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 6
Find:
∫sin5sin2d
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 7
Find:
∫sin5sin4d
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 8
Find:
∫sin6sin4d Source: JLS480(18)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 9
Find:
∫sin7sin3d Source: AM292(9)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 10
Find:
∫sin8sin3d
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Section II
SineCosine
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 11
Find:
∫sincos2 d
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 12
Find:
∫sin(32 ) cos 2 d
Source: PV401(25)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 13
Find:
∫sin2cosd Source: LC231(7)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 14
Find:
∫sin3cosd
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 15
Find:
∫sin4cos3d Source: RLR61(v)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 16
Find:
∫sin5cos3d
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 17
Find:
∫sin6cos2d Source: JLS479
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 18
Find:
∫sin7cos3d Source: AM292(8)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Section III
CosineSine
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 19
Find:
∫cos3sin4 d
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 20
Find:
∫cos2sind Source: AET355(15a)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 21
Find:
∫cos(52 ) sin 2 d
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 22
Find:
∫cos3sin2d Source: AET353(4), JG57(2ii)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 23
Find:
∫cos4sin2d Source: AM302(40)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 24
Find:
∫cos4sin3d Source: AET355(15b)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 25
Find:
∫cos5sin2d Source: DDB425(10)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 26
Find:
∫cos5sin3d Source: AM296(11)
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 27
Find:
∫cos5sin4d Source: PV380(21)
d f d l
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
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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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Section IV
CosineCosine
I d fi it I t ti U i P d t t S F l
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 29
Find:
∫coscos2 d
Indefinite Integration Using Product to Sum Formulae
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 30
Find:
∫cos2cosd Source: RLR63(xv)
Indefinite Integration Using Product to Sum Formulae
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Indefinite Integration Using Product-to-Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 31
Find:
∫cos3cos2d Source: AM302(41)
Indefinite Integration Using Product-to-Sum Formulae
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Indefinite Integration Using Product to Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 32
Find:
∫cos4cosd Source: PV380(22)
Indefinite Integration Using Product-to-Sum Formulae
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Indefinite Integration Using Product to Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 33
Find:
∫cos4cos2d Source: AM296(12)
Indefinite Integration Using Product-to-Sum Formulae
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Indefinite Integration Using Product to Sum Formulae
Questions compiled by Dr Lee Chu Keong
Question 34
Find:
∫cos5cos3d Source: DDB426(29), JLS481(19)
Indefinite Integration Using Product-to-Sum Formulae
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g g
Questions compiled by Dr Lee Chu Keong
Question 35
Find:
∫cos7cos3d Source: AM292(10)
Indefinite Integration Using Product-to-Sum Formulae
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Questions compiled by Dr Lee Chu Keong
Question 36
Find:
∫cos7cos5d
Indefinite Integration Using Product-to-Sum Formulae
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Questions compiled by Dr Lee Chu Keong
Question 37
Find:
∫cos15cos4d
Indefinite Integration Using Product-to-Sum Formulae
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Questions compiled by Dr Lee Chu Keong
Section V
Miscellaneous
Indefinite Integration Using Product-to-Sum Formulae
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Questions compiled by Dr Lee Chu Keong
Question 38
Find:
∫sinsin2sin3d
Indefinite Integration Using Product-to-Sum Formulae
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Questions compiled by Dr Lee Chu Keong
Question 39
Find:
∫sinsin3sin5d
Indefinite Integration Using Product-to-Sum Formulae
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Questions compiled by Dr Lee Chu Keong
Question 40
Find:
∫coscos2sin4d
Indefinite Integration Using Product-to-Sum Formulae
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Questions compiled by Dr Lee Chu Keong
Question 41
Find:
∫sin2cos5sec4 d
Indefinite Integration Using Product-to-Sum Formulae
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Questions compiled by Dr Lee Chu Keong
Question 42
Find:
∫cos sin8d
Indefinite Integration Using Product-to-Sum Formulae
i
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Questions compiled by Dr Lee Chu Keong
Question 43
Find:
∫sin cos4d Source: AET355(15c)
Indefinite Integration Using Product-to-Sum Formulae
Q i 44
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Questions compiled by Dr Lee Chu Keong
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
About Dr Lee Chu Keong
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About Dr Lee Chu Keong
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.