Indefinite Integration - Pure Trigometric Functions (B) - Questions

8/17/2019 Indefinite Integration - Pure Trigometric Functions (B) - Questions

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The Nail It Series Indefinite Integration of

“Pure” Trigonometric Functions (B)

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 12, 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 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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Indefinite Integration of “Pure” Trigonometric Functions

Questions compiled by Dr Lee Chu Keong

Question 1

Find:

∫sin5d Source: SIG323(2)


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Question 2

Find:

∫sin2d Source: AM198(7b), JMAW271(41)


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Question 3

Find:

∫sin2d Source: RCS136, WFO239(22), JH135(3v)


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Question 4

Find:

∫sin3d Source: TKS265(1i)


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Question 5

Find:

∫sin4d Source: RCS138(1ai), JMAW270(7)


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Question 6

Find:

∫sin5d Source: TKS265(2i), EWS374(1)


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Question 7

Find:

∫sind Source: WFO239(6)


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Question 8

Find:

∫sin + 1 d Source: TKS266(4iii)


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Question 9

Find:

∫sin2 + 1 d Source: TKS266(4i)


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Question 10

Find:

∫sin2 − 3 d Source: MW187(12)


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Question 11

Find:

∫sin3 − 1 d Source: RCS136(1c)


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Question 12

Find:

∫sin3 + 2 d


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Question 13

Find:

∫sin4 − 4 d

Source: TKS266(4iv)


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Question 14

Find:

∫sin + d Source: WFO239(8)


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Question 15

Find:

∫sin 2d Source: DDB425(1), RIP289(10a)


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Question 16

Find:

∫sin 3d Source: RCS136(24), MW195(5)


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Question 17

Find:

∫sin 7d Source: GM121(26)


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Question 18

Find:

∫sin 2d Source: PV401(6), DDB419(28), AM301(31)


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Question 19

Find:

∫sin 5d Source: PV380(2)


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Question 20

Find:

∫sin 2d Source: DDB421(3)


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Question 21

Find:

∫sin + sin 2 + sin 3 d Source: TKS265(2iv)


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Question 22

Find:

∫cos2 d Source: RCS136, WFO239(7)


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Question 23

Find:

∫cos2d Source: SIG322(1), SIG323(1), TKS265(2ii)


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Question 24

Find:

∫cos3d Source: RCS135, AM202(24), DDB419(1), WFO239(23)


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Question 25

Find:

∫cos4d Source: JMAW2708(a), EWS376(1)


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Question 26

Find:

∫ sin 4tan4 d Source: EWS399(61)

What is this question doing here?


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Question 27

Find:

∫cos2 + 3 d Source: RCS136(1d)


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Question 28

Find:

∫cos3 − 2 d Source: TKS266(4ii)


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Question 29

Find:

∫cos4 + 1 d Source: GM120(8)


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Question 30

Find:

∫cos7 + 5 d Source: TFWG326(6)


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Question 31

Find:

∫cos1 − 3 d Source: JH135(3vi)

d f f “ ”


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Question 32

Find:

∫ cos − 2 d Source: TKS266(4v)

I d fi it I t ti f “P ” T i t i F ti


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Question 33

Find:

∫12 cos(12 +

12) d

Source: RCS138(1ii)

I d fi it I t ti f “P ” T i t i F ti


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Question 34

Find:

∫cos5 − 3 d Source: EWS399(51)



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Indefinite Integration of Pure Trigonometric Functions


Question 35

Find:

∫cos 3d Source: LC262(8), TKS265(5iii), RIP236(33), AM294(2)



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Question 36

Find:

∫cos 4d Source: GM121(25)



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Question 37

Find:

∫cos 3 d Source: AM295(6)



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Question 38

Find:

∫cos 5d Source: EP463(27)



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Question 39

Find:

∫cos 2 d Source: PV401(26), AM301(33)



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g g


Question 40

Find:

∫cos2 d Source: MW195(2)



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Question 41

Find:

∫cos5 d Source: MW195(6)



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Question 42

Find:

∫cos6 2 d Source: AM301(35)



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Question 43

Find:

∫cos6 4d Source: EP462(12)



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Question 44

Find:

∫3 sin + 2 cos d Source: TKS264(4i)



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Question 45

Find:

∫(2 sin 3 + 12 cos2)d Source: TKS264(4iii)



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Question 46

Find:

∫2 cos 3 − sin 5 d Source: TKS265(1iv)



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Question 47

Find:

∫sin 4 − sin d Source: TKS265(2v)



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Question 48

Find:

∫tan2 d Source: WFO240(35)


i


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Question 49

Find:

∫tan 2 d Source: LC262(9), JLS480(11)


Q i 50


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Question 50

Find:

∫tan 4d Source: GM120(10)


Q ti 51


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Question 51

Find:

∫tan6 2d Source: PV380(8)


Q ti 52


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Question 52

Find:

∫csc2d Source: JLS480(9)


Q ti 53


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Question 53

Find:

∫csc 2d Source: AM204(59)


Q estion 54


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Question 54

Find:

∫csc 3d Source: RIP236(23)


Question 55


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Question 55

Find:

∫csc 2d Source: DDB419(26), AM302(54)


Question 56


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Question 56

Find:

∫csc 3d Source: PV380(24)


Question 57


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Question 57

Find:

∫csc6 2d Source: EP463(19)


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Question 59


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Question 59

Find:

∫sec3d Source: JLS480(10)


Question 60


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Question 60

Find:

∫sec 2 d Source: TKS265(1v)


Question 61


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Question 61

Find:

∫sec (37 ) d Source: TKS265(2iii)


Question 62


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Question 62

Find:

∫sec 2d Source: DDB419(2)


Question 63


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Question 63

Find:

∫ 1cos 2 d

What is this question doing here?Source: TFWG327(8)


Question 64


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Question 64

Find:

∫sec 5d Source: TKS265(1iii), EWS376(2)


Question 65


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Question 65

Find:

∫sec √ 2d Source: TKS265(2vi)


Question 66


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Question 66

Find:

∫sec2 + 1 d Source: JH135(3vii)


Question 67


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Quest o 6

Find:

∫sec3 + 2 d Source: TFWG328(19)


Question 68


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Q

Find:

∫sec4 + 5 d Source: TKS266(4vi)


Question 69


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Q

Find:

∫sec5 + 2 d Source: MW187(13)


Question 70


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Q

Find:

∫sec 2d Source: AM298(18)


Question 71


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Q

Find:

∫sec 7d Source: PV380(25)


Question 72


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Find:

∫sec2 − 1 d Source: DDB425(10)


Question 73


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Find:

∫sec6 2d Source: EP461(5)


Question 74


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Find:

∫cot2d Source: WFO239(26)


Question 75


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Find:

∫cot7d Source: MW187(9)


Question 76


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Find:

∫cot8 d Source: SIG323(10)


Question 77


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Find:

∫cot2 − d Source: SIG323(11)


Question 78


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Find:

∫cot 2d Source: LC232(9), PV401(2)


Question 79


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Find:

∫cot 2d Source: PV380(23), DDB447(64)


Question 80


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Find:

∫cot 3d Source: AM298(21), EP463(29)


Question 81


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Find:

∫cot6 4d Source: PV380(26)

Sources

Sources

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


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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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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.

Indefinite Integration - Pure Trigometric Functions (B) - Questions

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