indefinite integration - functions sine & cosine - questions

8/17/2019 Indefinite Integration - Functions Sine & Cosine - Questions

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The Nail It Series Indefinite Integration of

Functions Involving Sine & Cosine

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 at least fifty questions. The ideabehind 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 questions 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 very difficult to locate. As Winston Churchill oncesaid, “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 minimum of fifty questions, with each question appearing

on its own page. View it on your tablet or a mobile phone, 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 does

the same, but with two or more questions that look alike, but that require differentapproaches 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

About Learning

Many teachers today like to tell their students that learning is enjoyable, and that learning is

fun. What students quickly realise is that learning is often repetitive (and therefore boring),cognitively demanding (and therefore tiring), and time-consuming (and therefore costly). I’d

like to point out seven things that are needed for effective learning to take place. I suspect

teachers don’t mention them any longer because they are unpopular.

1. Learning takes hard work – a lot of hard work. But I’ve realised that all of life’s

worthwhile goals – setting up a business, starting a family, etc. can only be achieved with

hard work.

2. Learning takes dedication. There are no short cuts to learning. Learning is an intense

activity. Are you willing to learn at all cost?

3. Learning takes commitment . There are thing that you’ve going to have to give up, if you

want to learn. The price for mastering a subject matter is high. Are you willing to pay the

price (e.g., reducing the amount of time watching YouTube videos, or playing your

favourite computer game)?

4. Learning takes discipline. Closely tied to discipline is sacrifice, and a conscious effort to

minimise distractions. Are you willing to sacrifice (not meeting your friends so often,

watching less movies, etc.) in order to learn?

5. Learning takes motivation. And here, you have decide what exactly, motivates you. Are

you after an “A” grade, or are you after complete mastery of the subject matter? In other


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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 a destination that’s full of fun. 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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Integration of Trigonometric Functions Involving Only Sine and Cosine

Questions compiled by Dr Lee Chu Keong

Section I

Indefinite Integration of:

∫ sin cos d


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Question 1

Find:

∫sincosd


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Question 2

Find:

∫sincos d


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Question 3

Find:

∫ sincos d Source: AM202(25)


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Question 4

Find:

∫sincos d


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Question 5

Find:

∫sincos4 d Source: AM204(62)


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Question 6

Find:

∫sincos d


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Question 7

Find:

∫sincos6 d


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Question 8

Find:

∫sin √ cosd


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Question 9

Find:

∫sin √ cos d


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Section II


∫sin cos d


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Question 10

Find:

∫sincosd


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Question 11

Find:

∫sin cosd Source: AM200(15)


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Question 12

Find:

∫sin cosd


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Question 14

Find:

∫sin cosd [Porter, Examples 17a, Question 3]


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Question 15

Find:

∫sin6 cosd


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Question 16

Find:

∫ √ sincosd


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Question 17

Find:

∫ √ sin cosd


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Pause and Think

Study the solutions to Question 43 and 52. Are both

solutions really the same?


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Section III


∫sin ⋅ cos (where = )


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Integration of Trigonometric Functions I


Question 18

Find:

∫sincosd


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Question 19

Find:

∫sin cos d Source: AM296(8), MW194(2)


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Question 20

Find:

∫sin cos d Source: MW194(3)


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Question 21

Find:

∫sin4 cos4 d


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Question 22

Find:

∫sin3cos3d


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Question 23

Find:

∫sin 2cos 2d


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Question 24

Find:

∫ 1sincos d


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Question 25

Find:

∫ 1sin cos d


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Question 26

Find:

∫ cos2sin cos d


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Section IVa


∫ sin ⋅ cos (where is odd)

f


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Question 27

Find:

∫ cos

sin d

I i f T i i F i I


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Question 28

Find:

∫sin cos d Source: AM295(4)Note: = 3 (odd)

I t ti f T i t i F ti I


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Question 29

Find:

∫sin4 cos d Note: = 3 (odd)


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Question 31

Find:

∫ √ sincos d Source: PV402(32)



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Question 32

Find:

∫sin cos d Note: = 5 (odd)Source: DDB426(18)



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Question 33

Find:

∫sin4 cos d Note:

= 5 (odd)



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Question 34

Find:

∫ cos

sin d



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g g


Question 35

Find:

∫sin6 cos7 d


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Question 37

Find:

∫ cos

√ sin d Source: MW194(7)



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Question 38

Find:

∫ cos sin d Source: RIT1(6)



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Section IVb


∫ sin ⋅ cos (where is odd)



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Question 39

Find:

∫sin cos d



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Question 40

Find:

∫sin cos4 d



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Question 41

Find:

∫sin √ cosd



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Question 42

Find:

∫ sin

√ cos d



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Question 43

Find:

∫ sin

cos4 d


Q i 44


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Question 44

Find:

∫sin(2 + 3) cos(2 + 3) d Source: MW194(10)


Q i 45


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Question 45

Find:

∫sin cos d


Q ti 46


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Question 46

Find:

∫ sin

cos d


Q ti 47


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Question 47

Find:

∫ sincos d Source: AM202(25)


Q ti 48


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Question 48

Find:

∫ sin

√ cos d Source: MW194(8)



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Section IVc


∫ sin ⋅ cos (where and are both even)


Question 49


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Question 49

Find:

∫sin cos4 d


Question 50


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Question 50

Find:

∫sin4 cos d


Question 51


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Question 51

Find:

∫sin4 3cos 3d Source: AM296(9)


Question 52


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Question 52

Find:

∫cos 2sin4 2 d

Source: DDB422(5)


Question 53


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Question 53

Find:

∫ cos

sin6 d



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Section IVd


∫ sin ⋅ cos (where and are both − ve)


Question 54


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Question 54

Find:

∫ 1sincos d


Question 55


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Question 55

Find:

∫ 1sin cos d


Question 56


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Question 56

Find:

∫ 1sin cos d


Question 57


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Question 57

Find:

∫ sin

1 + cos d



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Section V


∫(sin + cos ) d (and variations)

Integration of “Pure” Trigonometric Functions

Question 58


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Quest o 58

Find:

∫(sin + cos ) d


Question 59


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Q

Find:

∫(sin + cos ) d


Question 60


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Q

Find:

∫(sin + cos ) d


Question 61


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Q

Find:

∫(sin4 + cos4 ) d


Question 62


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Q

Find:

∫(sin6 + cos6 ) d


Question 63


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Find:

∫(sin8 + cos8 ) d


Question 64


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Find:

∫(sin − cos ) d


Question 65


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Find:

∫ 1sin + cos d


Question 66


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Find:

∫ 1sin4 + cos4 d

Sources

Sources

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


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y , , , ( ) ( )

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.

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

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

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.

JMAW Marsden, J., & Weinstein, A. (1985). Calculus I . New York: Springer-Verlag.PV Purcell, E.J., & Varberg, D. (1987). Calculus with Analytic Geometry (5th ed.). Englewood Cliffs, NJ: Prentice-Hall.

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.

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.

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

About Dr Lee Chu Keong

About Dr Lee Chu Keong

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


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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 after

year 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 and India. 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 a

lot harder to find. For this reason, he started the Nail It Series, a series of ebooks containingquestions 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.

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