A Calculus Refresher

Transcription

1 A Calculus Refresher v. March 003 This work is licensed under the Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License. To view a copy of this license, visit or send a letter to Creative Commons, 7 Second Street, Suite 300, San Francisco, California, 9405, USA. c 003 mathcentre

2 Contents Foreword Preliminary work Howtousethisbooklet Reminders 3 Tables of derivatives and integrals 4. Derivatives of basic functions 5. Linearity in differentiation 7 3. Higher derivatives 9 4. The product rule for differentiation 0 5. The quotient rule for differentiation 6. The chain rule for differentiation 3 7. Differentiation of functions defined implicitly 5 8. Differentiation of functions defined parametrically 6 9. Miscellaneous differentiation exercises 7 0. Integrals of basic functions 0. Linearity in integration. Evaluating definite integrals 3 3. Integration by parts 4 4. Integration by substitution 6 5. Integration using partial fractions 9 6. Integration using trigonometrical identities Miscellaneous integration exercises 35 Answers 39 Acknowledgements 46

3 Foreword The material in this refresher course has been designed to enable you to cope better with your university mathematics programme. When your programme starts you will find that the ability to differentiate and integrate confidently will be invaluable. We think that this is so importantthatwearemakingthiscourseavailableforyoutoworkthrough eitherbeforeyoucometouniversity,orduringtheearlystagesofyour programme. Preliminary work You are advised to work through the companion booklet An Algebra Refresher before embarking upon this calculus revision course. Howtousethisbooklet Youareadvisedtoworkthrougheachsectioninthisbookletinorder. YoumayneedtorevisesometopicsbylookingatanAS-levelor A-level textbook which contains information about differentiation and integration. You should attempt a range of questions from each section, and check youranswerswiththoseatthebackofthebooklet.themorequestions that you attempt, the more familiar you will become with these vital topics.wehaveleftsufficientspaceinthebookletsothatyoucando any necessary working within it. So, treat this as a work-book. Ifyougetquestionswrongyoushouldrevisethematerialandtryagain until you are getting the majority of questions correct. Ifyoucannotsortoutyourdifficulties,donotworryaboutthis. Your university will make provision to help you with your problems. This may take the form of special revision lectures, self-study revision material or a drop-in mathematics support centre. Level This material has been prepared for students who have completed an A-level course in mathematics

4 Reminders Use this page to note topics and questions which you found difficult. Seek help with these from your tutor or from other university support services as soon as possible. 3

5 Tables The following tables of common derivatives and integrals are provided for revision purposes. It will be a great advantage to know these derivatives and integrals because they are required so frequently in mathematics courses. Table of derivatives f(x) x n ln kx e kx a x sin kx coskx tan kx f (x) nx n x ke kx a x ln a k coskx k sin kx k sec kx Table of integrals f(x) f(x) dx x n (n ) x = x x n+ n + + c ln x + c e kx e kx (k 0) k + c sin kx cos kx (k 0) + c k coskx (k 0) sin kx + c k sec kx (k 0) tan kx + c k 4

6 . Derivatives of basic functions Try to find all the derivatives in this section without referring to a table of derivatives. The derivatives of these functions occur so frequently that you should try to memorise theappropriaterules.ifyouarereallystuck,consultthetableonpage4.. Differentiate each of the following with respect to x. (a) x (b) x 6 (c) 6 (d) x (e) x (f) x /7 (g) x 3 (h) x 79 (i) x.3 (j) 3 x (k) x 5/3 (l) x 0.7. Differentiate each of the following with respect to θ. (a) cosθ (b) cos 4θ (c) sin θ (d) sin θ 3 (g) sin( 8θ) (h) tan θ ( (i) cos 3πθ (j) cos 5θ ) 4 (e) tan θ (k) sin 0.7θ (f) tanπθ 5

7 3. Find the following derivatives. (a) d dx (ex ) (b) d dy (ey ) (c) d dt (e 7t ) (d) d dx (e x/3 ) (e) d dz (ez/π ) (f) d dx (e.4x ) (g) d dx (3x ) 4. Find the following derivatives. (a) d dx (ln x) (b) d dz (ln 5z) (c) d ( ln x ) dx 3 6

8 . Linearity in differentiation The linearity rules enable us to differentiate sums and differences of functions, and constant multiples of functions. Specifically d d (f(x) ± g(x)) = dx dx (f(x)) ± d dx (g(x)),. Differentiate each of the following with respect to x. d dx (kf(x)) = k d dx (f(x)). (a) 3x + (b) x x (c) cos x sin x (d) 3x sin 4x (e) e x + e x (f) x 4 3 lnx (g) 4x5 3 tan8x e 5x. Find the following derivatives. (a) d ( ) 5t /5 + t8 (b) d ( cos θ4 ) dt 8 dθ 3e θ/4 (c) d dx ( ) 3e 3x/5 5 ( 9 tan 3x 3 4 cos 8x ) ( 4 z4/3 3 e 4z/3 ) (d) d dx (e) d dz 7

9 InQuestions 3-5youdon t need the product rule, quotient rule or chain rule to differentiateanyoftheseifyoudothealgebrafirst! 3. Expand the powers or roots and hence find the following derivatives. (a) d dy ( ) y (b) d dx ( (x) 3 (x) 3 ) (c) d dy ( ( ey ) 4 ) (d) d dt ( 3 5e t) 4. Simplify or expand each of the following expressions, and then differentiate with respect to x. (a) x x (b) x( ( x x ) (c) x ) ( ) 3 x 3 x x + x (d) (e x )(3 e 3x ) (e) e x e 4x 5. Use the laws of logarithms to find the following derivatives. (a) d ( ) ln x 9/ (b) d ( ( )) ln (c) d ( ( )) t 3 ln dx dx 6x dt e 3t (d) d dt (ln ( te t) /3 ) 8

10 . Find the following second derivatives. 3. Higher derivatives (a) d dx (x5 ) (d) d dy(8 3y) ( (g) d x 3/ ) dx x 3/ (b) d dx(cos 3x) ( ) d (e) dx x 3x 3x3 (h) d dx (ex + e x + sin x + cosx) d (c) e z ) dz (ez (f) d dt (ln t 6t) ( (i) d dt sin t ) 4 ln 4t 9

11 4. The product rule for differentiation The rule for differentiating the product of two functions f(x) and g(x) is d dx (f(x)g(x)) = f (x)g(x) + f(x)g (x).. Differentiate each of the following with respect to x. (a) x sin x (b) x 3 cos x (c) x /3 e 3x (d) x ln 4x (e) (x x) sin 6x (f) x ( tan x 3 cos x ) 3. Find the following derivatives. (a) d dθ (sin θ cosθ) (b) d dt (sin t tan 5t) (c) d dz (sin z ln 4z) (d) d dx ( e x/ cos x ) (e) d dx (e6x ln 6x) (f) d dθ (cosθ cos 3θ) (g) d (lnt ln t) dt 0

12 5. The quotient rule for differentiation The rule for differentiating the quotient of two functions f(x) and g(x) is ( ) d f(x) = f (x)g(x) f(x)g (x). dx g(x) (g(x)). Differentiate each of the following with respect to x. (a) x x (b) x 4 x (c) x + x (d) 3x x 3 x (e) + x x x

13 . Find the following derivatives. (a) d ( ) sin x (b) d ( ) ln x dx x dx x 4/3 (c) d ( ) θ dθ tan θ (d) d dz ( e z z ) (e) d ( ) x dx ln x 3. Find the following derivatives. (a) d ( ) sin t (b) d ( ) e x dt sin 5t dx tan x (c) d ( ) ln x dx cos 3x (d) d dx ( ) ln 3x ln 4x

14 6. The chain rule for differentiation Thechainruleisusedtodifferentiatea functionofafunction : d dx (f(g(x))) = f (g(x)).g (x).. Differentiate each of the following with respect to x. (a) (4 + 3x) (b) ( x 4 ) 3 (c) (e) ( x x) /3 (f) (x 3x + 5) 5/ (g) x x ( x) (d) + x (h) 4x x 4 3

15 . Find the following derivatives. (Remember the notation for powers of trigonometricfunctions: sin x means (sin x),etc.) (a) d dθ (sin θ) (b) d dθ (sin θ ) (c) d dθ (sin(sin θ)) (d) d (tan(3 4x)) dx (e) d dz (cos5 5z) (f) d ( ) dx cos 3 x (g) d dt (sin( t 3t )) 3. Findthefollowingderivatives. (Thenotation exp x isusedratherthan e x where it is clearer.) (a) d dy ( exp( y ) ) (b) d (exp(cos 3x)) dx (c) (e) d dx (sin(ln 4x)) (f) d dx (ln(ex e x )) (g) d dt d dx (cos(e3x )) ( e 3t 3 cos 3t ) (d) d (ln(sin 4x)) dx 4

16 7. Differentiation of functions defined implicitly. Find dy intermsof ywhen xand yarerelatedbythefollowingequations. You dx willneedtheformula dy = dx /dx. dy (a) x = y y 3 (b) x = y + y (c) x = e y + e y (d) x = ln(y e y ). Find dy dx equations. intermsof xand/or y when xand y arerelatedbythefollowing (a) cos x = tany (b) x + y = y x (c) y sin y = cos x (d) e x x = e y + y (e) x + e y = ln x + ln y (f) y = (x y) 3 5

17 8. Differentiation of functions defined parametrically If xand yarebothfunctionsofaparameter t,then. Find dy dx equations. dy dx = dy dt dx dt intermsof twhen xand yarerelatedbythefollowingpairsofparametric (a) x = sin t, y = cost (b) x = t t, y = t (c) x = e t + t, y = e t + t (d) x = ln t + t, y = t ln t. Find dx dy equations. intermsof twhen xand yarerelatedbythefollowingpairsofparametric (a) x = 3t + t 3, y = t + t 4 (b) x = cos t, y = tan t 6

18 9. Miscellaneous differentiation exercises. Find the following derivatives, each of which requires one of the techniques covered in previous sections. You have to decide which technique is required for each derivative! (d) (a) d dx (x3 tan 4x) (b) d dt (tan3 4t) (c) d (exp(3 tan4x)) dx (e) d dx (exp(x ex )) (f) d ( y 4 + y 4 ) dy y + y (g) d dx (x x ) (i) d ( ) 5 3x dx (j) d dt (ln(ln t)) (k) d ( ( ) z ) ln dz + z d ( ) 3θ dθ tan4θ (h) d ( ) dx ln x x. Find the following derivatives, which require both the product and quotient rules. (a) d ( ) x cosx (b) d ( e z ) (c) d ( ) sin 3θ cos θ dx cosx dz z ln z dθ tan4θ 7

19 3. Find the following derivatives, which require the chain rule as well as either the product rule or the quotient rule. (a) d dt (e t ln(e t + )) (b) d dθ (sin 3θ cos 4 3θ) (c) d ( ) x 3/ dx + x (d) d (exp(θ cosθ)) dθ (e) (g) d ( ) dy y y d dx ( ) (x ln x) 3 (f) d ( exp dx ( )) x + x 4. Find the following derivatives, which require use of the chain rule more than once. (a) d dx ( cos 3 x) (b) d ( ( exp (x x ) /4)) (c) d ( ( ln tan )) dx dθ θ 8

20 5. Find the following second derivatives. (a) d dx ( + x ) (b) d dz (exp(z )) (c) d dθ (sin3 θ) ( ) (d) d dx ( x 4 ) 4 6. Rememberingthatcosec x = sin x, sec x = cos x following derivatives. and cot x = cosx sin x,findthe (a) d dx (cosec x) (b) d dθ (sec θ) (c) d dz ( + cot z) (d) d dθ (cosec θ cot 3 θ) (e) d dx (ln(sec x + tanx)) (f) d (tan(sec θ)) dθ 9

21 0. Integrals of basic functions Trytofindalltheintegralsinthissectionwithoutreferringtoatableofintegrals. The integrals of these functions occur so frequently that you should try to memorise theappropriaterules.ifyouarereallystuck,consultthetablesonpage4.. Integrate each of the following with respect to x. (a) x 4 (b) x 7 (c) x / (d) x /3 (e) x (f) x / (g) 4 x (h) x 3 (i) x 0. (j) x 0.3 (k) x (l) x 3 (m) x (n) x 4/3. Integrate each of the following with respect to x. (a) cos 5x (b) sin x (c) sin x (d) cos x (e) x (f) e x (g) e x (h) e x/3 (i) e 0.5x (j) e x (k) e x (l) cos( 7x) 0

22 . Linearity in integration The linearity rules enable us to integrate sums(and differences) of functions, and constant multiples of functions. Specifically (f(x) ± g(x))dx = f(x) dx ± g(x)dx, k f(x)dx = k f(x)dx. Integrate each of the following with respect to x. (a) 7x 4 (b) 4x 7 (c) x / + x /3 (d) 7x /3 (e) x x (f) x + x (g) x 3 + x (h) 7x 3 (i) (j) x 0.3 (k) x x (l) 7x. Integrate each of the following with respect to x. (a) 3x + cos 4x (b) 4 + sin 3x (c) x + sin x (d) 4e x + cos x (e) e x + e x (f) 3 sin x + sin 3x (g) kx, kconstant (h) 4 x (i) + x + x (j) 3x 7 (k) cos x (l) x 3x /

23 3. Simplify each of the following expressions first and then integrate them with respect to x. (a) 6x(x + ) (b) (x + )(x ) (c) x3 + x x (d) ( x + )( x 3) (e) e x (e x e x ) (f) e3x e x (g) x + 4 e x x (h) x + 3x + x +

24 . Evaluating definite integrals. Evaluate each of the following definite integrals. (a) (e) (i) 0 3 / 0 7x 4 dx (b) (s + 8s 3 )ds (f) e 3x dx (j) t 7 dt (c) (x / + x /3 )dx (d) 3 x dx (g) (t + t)dt (h) 0 π/4 dx (k) (λ + sin λ)dλ (l) e x 0 π/ t /3 dt cos x dx (e x + e x )dx. Forthefunction f(x) = x + 3x verifythat 0 f(x)dx + 3 f(x)dx = 3 0 f(x)dx 3. Forthefunction f(x) = 4x 7xverifythat f(x)dx = f(x)dx. 3

25 3. Integration by parts Integrationbypartsisatechniquewhichcanoftenbeusedtointegrateproductsof functions.if uand varebothfunctionsof xthen u dv dx dx = uv v du dx dx When dealing with definite integrals the relevant formula is b u dv b a dx dx = [uv]b a v du a dx dx. Integrate each of the following with respect to x. (a) xe x (b) 5x cosx (c) x sin x (d) x cos x (e) x ln x (f) x sin x. Evaluate the following definite integrals. (a) π/ 0 x cosxdx (b) 4xe x dx (c) 5te t dt (d) 3 x ln x dx 3. In the following exercises it may be necessary to apply the integration by parts formula more than once. Integrate each of the following with respect to x. (a) x e x (b) 5x cosx (c) x(ln x) (d) x 3 e x (e) x sin x 4

26 4. Evaluate the following definite integrals. (a) π/ 0 x cosxdx (b) 0 7x e x dx (c) t e t dt 5. Bywriting ln xas ln xfind ln x dx. 6. Let Istandfortheintegral lnt t dt. Using integration by parts show that I = (lnt) I (plusaconstantofintegration ) Hencededucethat I = (ln t) + c. 7. Let Istandfortheintegral e t sin t dt. Usingintegrationbypartstwiceshow that I = e t sin t e t cost I (plusaconstantofintegration ) Hence deduce that e t sin t dt = et (sin t cost) + c 5

27 4. Integration by substitution. Find each of the following integrals using the given substitution. (a) cos(x 3)dx, u = x 3 (b) sin(x + 4)dx, u = x + 4 (c) cos(ωt + φ)dt, u = ωt + φ (d) e 9x 7 dx, u = 9x 7 (e) x(3x + 8) 3 dx, u = 3x + 8 (f) x 4x 3 dx, u = 4x 3 (g) (x ) 4dx, u = x (h) (3 t) dt, u = 3 t 5 (i) xe x dx, u = x (j) sin x cos 3 x dx, u = cosx (k) t dt, u = + t (l) + t x dx, u = x + x + 6

28 . Find each of the following integrals using the given substitution. x (a) dx, z = x 3 (b) (x 5) 4 (x + 3) dx, u = x 5 x 3 3. Evaluate each of the following definite integrals using the given substitution. (a) (c) (d) (f) π/4 0 3 π π /4 cos(x π)dx, z = x π (b) 9 8 (x 8) 5 (x + ) dx, u = x 8 t t dt, byletting u = t,andalsobyletting u = t t(t + 5) 3 dt, u = t + 5 sin x x dx, u = x (e) (g) π/4 0 tanx sec x dx, u = tanx e t t dt, u = t 7

29 4. Bymeansofthesubstitution u = 4x 7x + showthat 8x 7 4x 7x + dx = ln 4x 7x + + c 5. Theresultofthepreviousexerciseisaparticularcaseofamoregeneralrule with which you should become familiar: when the integrand takes the form derivative of denominator denominator theintegralisthelogarithmofthedenominator. Usethisruletofindthefollowing integrals, checking each example by making an appropriate substitution. (a) x + dx (b) x 3 dx (c) 3 3x + 4 dx (d) x + dx 6. UsethetechniqueofQuestion5togetherwithalinearityruletofindthefollowing x integrals. For example, to find x 7 dxwenotethatthenumeratorcanbemade equal to the derivative of the denominator as follows: x x 7 dx = x x 7 dx = ln x 7 + c. x (a) x + dx (b) sin 3θ + cos 3θ dθ (c) 3e x dx + ex 8

30 5. Integration using partial fractions. By expressing the integrand as the sum of its partial fractions, find the following integrals. x + 3x + (a) x + x dx (b) x + x dx x + (d) ( x)(x ) dx (e) (g) 4x 4 x dx (h) 9x + 5 x + 0x + 9 dx 5x + 5 x + 7x + 0 dx (c) 5x + 6 x + 3x + dx (f) 5x x + 0x + 9 dx (i) ds s. By expressing the integrand as the sum of its partial fractions, find the following definite integrals. (a) 8x x + x dx (b) 0 5x + 7 (x + )(x + ) dx (c) 0 7x x 3x + dx 9

31 3. By expressing the integrand as the sum of its partial fractions, find the following integrals. x (a) x x + dx (b) 4x + 6 (x + ) dx (c) 7x 3 x 6x + 9 dx 4. By expressing the integrand as the sum of its partial fractions, find the following definite integrals. (a) 0 x + 8 x + 6x + 9 dx (b) x + 9 x + 8x + 8 dx (c) 0 8x x + x + dx 30

32 5. Find x + 6x + 5 (x + )(x + ) dx. 6. Find 5x + x 34 (x )(x 3)(x + 4) dx. 7. Find x x 4 (x + )(x + )(3 x) dx. 3

33 8. Inthisexamplenotethatthedegreeofthenumeratorisgreaterthanthedegree x 3 of the denominator. Find (x + )(x + ) dx. 9. Showthat x 3 + (x + )(x + ) canbewrittenintheform x (x + ) 9 x + Hence find x 3 + (x + )(x + ) dx. 0. Find 4x 3 + 0x + 4 dx by expressing the integrand in partial fractions. x + x 3

34 6. Integration using trigonometrical identities Trigonometrical identities can often be used to write an integrand in an alternative form which can then be integrated. Some identities which are particularly useful for integration are given in the table below. Table of trigonometric identities sin A sin B = (cos(a B) cos(a + B)) cosa cos B = (cos(a B) + cos(a + B)) sin A cosb = (sin(a + B) + sin(a B)) sin A cosa = sin A cos A = ( + cos A) sin A = ( cos A) sin A + cos A = tan A = sec A. In preparation for what follows find each of the following integrals. (a) sin 3x dx (b) cos 8x dx (c) sin 7t dt (d) cos 6x dx. Usetheidentity sin A = ( cos A)tofind sin x dx. 3. Usetheidentity sin A cosa = sin Atofind sin t costdt. 33

35 4. Find (a) sin 7t cos 3t dt (b) 8 cos 9x cos 4x dx (c) sin t sin 7t dt 5. Find tan t dt. (Hint:useoneoftheidentitiesandnotethatthederivativeof tan tis sec t). 6. Find cos 4 t dt.(hint:squareanidentityfor cos A). 7. (a)usethesubstitution u = cos xtoshowthat sin x cos n x dx = n + cosn+ x + c (b)inthisquestionyouarerequiredtofindtheintegral sin 5 t dt. Startbywriting theintegrandas sin 4 t sin t. Taketheidentity sin t = cos tandsquareitto produceanidentityfor sin 4 t. Finallyusetheresultinpart(a)tofindtherequired integral, sin 5 t dt. 34

36 7. Miscellaneous integration exercises To find the integrals in this section you will need to select an appropriate technique from any of the earlier techniques.. Find (9x ) 5 dx.. Find t + dt. 3. Find t 4 lntdt. 4. Find (5 t 3t 3 + ) dt. 35

37 5. Find (cos 3t + 3 sin t) dt. 6. Bytakinglogarithmstobaseeshowthat a x canbewrittenas e x lna.hencefind a x dxwhere aisaconstant. 7. Find xe 3x+ dx. 8. Find x (x + )(x ) dx. 36

38 9. Find tan θ sec θ dθ. 0. Find π/ 0 sin 3 x dx.. Usingthesubstitution x = sin θfind x dx.. Find e x sin x dx. 37

39 3. Find x 9 x(x )(x + 3) dx. 4. Find x 3 + x 0x 9 (x 3)(x + 3) dx. 5. Let I n standfortheintegral x n e x dx. Usingintegrationbypartsshowthat I n = xn e x n I n.thisresultisknownasareductionformula.useitrepeatedly tofind I 4,thatis x 4 e x dx. 6. Find 0 (4 t ) 3/dt,byletting t = sin θ. 38

40 Answers Section. Derivatives of basic functions. (a) (b) 6x 5 (c)0 (d) x (e) x (f) 7 x 6/7 (g) 3 x 4 (h) 79x 78 (i).3x 0.3 (j) 3 x 4/3 (k) 5 3 x 8/3 (l) 0.7 x.7. (a) sin θ (b) 4 sin 4θ (c) cos θ (d) cos θ 3 3 (e) sec θ (f) π sec πθ (g) 8 cos( 8θ) = 8 cos 8θ (h) 4 sec θ (i) 3π sin 3πθ (j) 5 sin ( ) 5θ 4 = 5 sin 5θ (k) 0.7 cos0.7θ 3. (a) e x (b) e y (c) 7e 7t (d) 3 e x/3 (e) π ez/π (f).4e.4x (g) 3 x ln 3 4. (a) x (b) z (c) x Section. Linearity in differentiation. a)3 b) x c) sin x cos x d) 9x cos4x e) e x e x f) x 3 x g) 0x 4 4 sec 8x 0e 5x. (a) t 4/5 + t 7 (b) sin θ e θ/4 (c) 9e3x/5 5 (d) 3 sec 3x + 6 sin 8x 3. (a) (e) 3 z/ e 4z/3 y (b) 4x + 3 (c) 8x 4 4 e4y (d) 3 3 5e t 4. (a) + x 3 x (b) 3 x 3x (c) x x x 4 (e) 4e 4x e x (d) 6e x 5e 5x + 3e 3x 5. (a) 9 x (b) x (c) 3 t 3 (d) 3t 3 Section 3. Higher derivatives. (a) 0x 3 (b) 9 cos 3x (c) 4e z 4e z (d)0 (e) x 3 8x (f) t t 3/ (g) 3 4 x / 5 4 x 7/ (h) e x + e x sin x cosx (i) sin t + 4t 39

41 Section 4. The product rule for differentiation. (a) sin x + x cosx (b) 3x cos x x 3 sin x (c) 3 x 4/3 e 3x 3x /3 e 3x (d) ln 4x + x x (e) (x ) sin 6x + 6(x x) cos 6x (f) x ( tan x 3 cos x 3. (a) cos θ sin θ (b) cos t tan5t + 5 sin t sec 5t (c) cosz ln 4z + sinz (d) ( z e x/ cos x + sin ) x (e) 6e 6x ln 6x + e6x x (g) (ln t + ln t) t (f) sin θ cos 3θ 3 cosθ sin 3θ Section 5. The quotient rule for differentiation ) ( + 3x sec x + sin ) x 3 3. (a) +x ( x ) (b) 4x3 3x 4 ( x) (c) 5 (d) 8x 8x 6x 4 (x 3 +3) (e) ( x + x (+x) ) /( x x) x cos x sinx. (a) (b) ( 4 ln x 3 x)x 7/3 (c) θ tan θ θ sec θ tan θ (d) ez( z z x lnx x ) (e) z (ln x) 3. (a) (c) cos t sin5t 5 cos 5t sint sin 5t (cos 3x)/x+3 lnx sin3x cos 3x, (d) ln(4/3) x(ln 4x) (b) e x tan x e x sec x tan x Section 6. The chain rule for differentiation. (a) 6(4 + 3x) (b) x 3 ( x 4 ) (c) 4 (d) x +x (e) 3 (g) ( ) x x x (h) ( x) 3 ( ) ( ) + x x 4/3(f) 5 (4x x 3)(x 3x + 5) 3/ 4x x3 (4x x 4 ) 3/. (a) sin θ cosθ (b) θ cosθ (c) cosθ cos(sin θ) (d) 4 sec (3 4x) (e) 5 cos 4 5z sin 5z (f) 3sin x cos 4 x (g) ( 6t) cos( t 3t ) 3. (a) y exp( y ) (b) 3 sin 3x exp(cos 3x) (c) 3e 3x sin(e 3x ) 4 cos 4x (d) sin4x (e) cos(ln 4x) x (f) ex +e x e x e x (g) 3e3t +9sin 3t e 3t 3cos 3t 40

42 Section 7. Differentiation of functions defined implicitly. (a) 3y (b) y y 3 (c) e y +e y (d) y e y +e y. (a) sin x sec y (b) x y (c) sinx cos y (d) ex (e) ( e y + x ) / ( ) e y y (f) 3(x y) +3(x y) Section 8. Differentiation of functions defined parametrically. (a) tan t (b) t3 t + (c) et +t e t + (d) t +t. (a) 3 4t (b) sint sec t Section 9. Miscellaneous differentiation exercises. (a) 3x tan 4x + 4x 3 sec 4x (b) tan 4t sec 4t (c) exp(3 tan4x) sec 4x (d) 3tan 4θ θ sec 4θ tan 4θ (g) ( x ln )x + x+ x (j) tlnt. (a) cos x cos x xsinx ( cos x) (e) ( e x ) exp(x e x )(f) 3y4 +5y 5y 4 3y 6 (y+y ) (i) ( 3 ln 5)5 3x (h) x x(ln x x) (k) 4 z (b) ez (z ln z lnz ) (z ln z) (c) (3cos 3θ cos θ sin3θ sinθ) tan 4θ 4sec 4θ sin 3θ cos θ tan 4θ 3. (a) e t ln(e t + ) + e t + (c) 6x( x ) / (+x ) 5/ (e) 3(x lnx) ( + ln x) (g) 3y y 3 (y ) 3/ (b) 6 sin 3θ cos 5 3θ sin 3 3θ cos 3 3θ (d) (cosθ θ sin θ) exp(θ cosθ) (f) exp ( ) x (+x) +x 4. (a) 3cos x sinx cos 3 x (b) 4 (x x ) 3/4 ( x) exp ( (x x ) /4) (c) sec (/θ) θ tan(/θ) 5. (a) (+x ) 3/ (b) ( + 4z ) exp(z ) (c) 6 sin θ cos θ 3 sin 3 θ (d) 30x6 ( x 4 ) x ( x 4 ) 5 6. (a) cot xcosec x (b) sec θ tan θ (c) cosec z +cot z (d) cosec θ cot 4 θ 3cosec 4 θ cot θ (e) sec x (f) sec (sec θ) sec θ tanθ 4

43 Section 0. Integrals of basic functions. (a) x5 5 + c, (b) x8 8 + c, (c) x3/ 3 + c (d) 3x4/3 4 + c, (e)sameas(c), (f) x / + c, (g) 4x5/4 + c, (h) + c, (i) x. x0.7 + c, (j) + c 5 x. 0.7 (k)sameas(f)(l) x / + c (m) x + c (n) 3x7/3 7 + c. (a) sin 5x + c, (b) 5 cos x + c (c) cos x + c (d) sin x + c (e) ln x + c (f) ex + c (g) e x + c (h) 3e x/3 + c (i) e 0.5x + c (j) e x + c (k)sameas(g) (l) sin( 7x) + c = sin 7x + c 7 7 Section. Linearity in integration. (a) 7x5 5 + c, (b) x8 + c, (c) x3/ 3 + 3x4/3 4 + c (d) 5x4/3 4 + c, (e) x3/ x / + c (f) x3 x4 + ln x + c (g) + c (h) + c x 4x (i) x + c (j) x c (k) x ln x + c (l) 7x x + c. (a) 3x + sin 4x + c (b) 4x x cos 3x + c (c) cos x + c (d) 4e x + sin x + c (g) k (e) e x + ex x cos 3x + c (f) 3cos + c 3 ln x + c (h) x 4 ln x + c (i) x + x + x3 3 + c (j) ln x 7x + c (k) sin x3 x + c (l) 3 6 6x/ + c 3. (a) x 3 + 3x + c (b) x3 x x7/ x + c (c) + 4x5/ + c (d) x x3/ 3 6x + c (e) e3x 3 ex + c (f) ex ex + c (g) x + 4 ln x + c (h) x + x + c Section. Evaluating definite integrals. (a) 7 (b) 6305 (c) (d) 5( 3 6) (e)68 4 (f) 4 (g)8 (h) (i) 5 3 (e3/ ) (j) (e e ) (k) π 6 + (l) e e f(x) dx = 33, f(x) dx = , f(x) dx = 8 3, f(x)dx = 8 3. f(x) dx = 7 6.Then = 33. 4

44 Section 3. Integration by parts. (a) xe x e x + c (b) 5 cosx + 5x sin x + c (c) sin x x cos x + c (d) cos x + x sin x + c (e) 4 x ln x 4 x + c (f) 8 sin x 4x cos x + c. (a) π (b) 4e (c) 5 4 e 5 4 e (d) 9 ln 3 3. (a) e x (x x + ) + c (b) 5x sin x 0 sinx + 0x cosx + c (c) x (ln x) x ln x + 4 x + c (e) x cos( x) + 6 cos( x) + 8x sin( x) + c 4. (a) π 4 (b) 7e 4 (c) 5 4 e + 4 e 5. x ln x x + c. Section 4. Integration by substitution (d) e x (x 3 + 3x + 6x + 6) + c. (a) sin(x 3) + c (b) cos(x + 4) + c (c) sin(ωt + φ) + c ω (d) 9 e9x 7 + c (e) 4 (3x + 8) 4 + c (f) (4x 3)5/ 40 + (4x 3)3/ 8 + c (g) 3(x ) 3 + c (h) 4(3 t) 4 + c (i) e x + c (j) cos4 x + c (k) ( + t ) / + c (l) x+ ln x + + c (x 3)3/. (a) + 6 x 3 + c 3 (x 5)7 8(x 5)6 64(x 5)5 (b) c whichcanbeexpandedto 7 x7 7 3 x x x 4 05 x x x + c 3 3. (a) (b) (c) 6 (d) (e) (f) (g) (e e ) 5. (a) ln x + + c (b) ln x 3 + c (c) ln 3x c (d) ln x + + c. 6. (a) ln x + + c (b) 3 ln + cos 3θ + c (c) 3 ln + ex + c 43

45 Section 5. Integration using partial fractions. (a) ln x + ln x + + c (b) ln x + + ln x + c (c) 4 ln x + + ln x + + c (e) 7 ln x ln x + + c (g) ln x + ln x + c (i) ln s ln s + + c. (a) 9 ln3 7 ln (b) ln3 + 3 ln (c) 3 ln3 ln (d) ln x 3 ln x + c (f) 7 ln x + 9 ln x + + c (h) 8 ln x ln x + + c 3. (a) ln x + c (b) 4 ln x + + c x x+ (c) 7 ln x c x 3 4. (a) ln 3 + ln + 5 (b) ln5 4 ln + 40 (c) ln3 5. ln x + + ln x + x+ + c. 6. ln x ln x 3 + ln x + c. 7. ln x 3 + ln x + ln x + + c. 8. x 3x ln x ln x + + c. 9. x ln x ln x + + c. x+ 0. x x + 4 ln x + 3 ln x + + c Section 6. Integration using trigonometrical identities. (a) 3 cos 3x + c (b) 8 sin 8x + c (c) 7 x. sinx + c cos t + c. 4 cos 7t + c (d) sin6x 6 + c 4. (a) cos 0t cos 4t + c (b) 4 4 sin 5x + sin 3x + c (c) sin 6t sin 8t + c 6 5. tant t + c t + sin t + sin 4t + c cost + 3 cos3 t 5 cos5 t + c. 44

46 Section 7. Miscellaneous integration exercises. (9x ) 6 + c. 54. t ln t + + c.(hint:let u = t +.) 3. 5 t5 ln t 5 t5 + c t3/ 3t4 + t + c sin 3t 3 cost + c lna exln a + c = ax + c. lna 7. e 3x+ ( x 3 9 ) + c 8. ln x + + ln x + c. 9. tan θ + c. 0. /3.. arcsin x + x x + c.. 5 ex (sin x cosx) + c 3. 3 ln x ln x ln x c. 4. x + x ln x ln x 3 + c ex [x 4 4x 3 + 6x 6x + 3] + c

47 Acknowledgements Thematerials inacalculus Refresherwere prepared by DrTony CroftandDr Anthony Kay, both at the Department of Mathematical Sciences, Loughborough University. TheauthorswouldliketoexpresstheirappreciationtoDrJoeKyleandJonathanDe Souza for their many helpful corrections and suggestions. This edition has been published by mathcentre in March 003. Users are invited to send any suggestions for improvement to Departments may wish to customise the latex source code of this booklet. Further details from: mathcentre c/o Mathematics Education Centre Loughborough University LE 3TU tel: www: 46