CALCULUSTHOMAS'CALCULUSARLYTRANSCENChapter 7JAMESSTEWARTIntegralsandTranscendental FunctionsERMStCIHeAA7.11.Integrals ofthetangent,cotangent,secant,andcosecantIntegrals ofTrigonometricfunctionsFunctions广TABLE 3.1 Derivatives of the inverse trigonometric functionsIntegrals of the tangent, cotangent, secant, and cosecant functionsdu/drd(sinw)]<1drVi-w d = In[sec u| + Csec ir dv = In see a + tan r| + Cd(cos-a)du/dx[]<1dVi-wcot u du In|sin arl + Cse r dr = In Jese w + cot a] + Cd(tan~lu)du/dx+ud[cotu du=-Injcsca+Cdu/drd(cor-a)If w is a diffierentiable furdI+nTZ0d(see""w)du/dr[Lau-Inul +C.[ >1(4)da/Ve-7d(cse"lw)du/dx(μ| > 1dxIu/V-1A
2016/11/15 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 7 Integrals and Transcendental Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.1 Integrals of Trigonometric Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley 1. Integrals of the tangent, cotangent, secant, and cosecant functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley cot ln csc u du u C
2ydy&rdIn(4)In(y-25)+C5H-CExereses243sec2fsecytanyh(2+see y)+CIn(6+3tant)+C 5.6+3tan/h62 + see)7.2seexdtnI+)+Cs..2 /n(sec x+tan x)+C2VT+2VIn(seex+tanx)21.+tcwttanwtdSeparableDifferential Equationse+not+dP22.AsineJarnrVhln(n)Zre cosfe'ydr[sine'-sin!23,Ze"cose"dyIn(1+e')+C26=0r=1-n(I+e)+C25.teLogn3Logiax20g0X+433837.x+1AJ10-ogz+Cin2-(og,t=2in2[n10-(og,(x+)=In10Slide4-8differentialEXAMPLE1Solve the differential equationtgiatogdyg(x)dydxh(y)=(I+ye, y>-1its differential form allows us to collect all y terms with dly and all x terms with drSinceI + y is never zero for ySolution1, we can solve theequation by separating(y)ay=g(x)dxthe variables.Now we simply integrate both sides of this equation:89黑-（+岁)dyg(x) dr(4)Treataridrasaountientotdiffesemtials md multiplydy=(1 + yledxboth silesi hy adr.After completing the integrations we obtain the solution y defined implicitly as a functionofx.=edDnidehylty1+ :[,-[ealetegrale borth sidesthoCEDIn(1+y)-e+ccoautama of istegration.The last equation gives y as an implicit fiunction of x.Slide 4-9Slide 4- 10SeparableDifferential EquationsExercisesSolve the differential equation in Exercises 9-22.dy层=+1)EXAMPLE2Solve the equation y(x + 1) -dyd10.-PVy>09.2Vxy>0-1.dSolutionWe change to differential form, separate the variables, and integrate:dy12.411.de=32e7dx(x + 1)dy =x(y2 + 1)da14. VE-Iydyad13.-VycosV,+*12+1x+1dx[-(-)9.2-1+c10. 2y -*+cDivider by # + 1.12.2=x+C11.e=e+cn(1+y)=x-nx+1|+c.14.c13.2tan o=x+cThe last equation gives the solution y as an implicit function of xSlide 4-11Slide 4.12
2016/11/15 2 Exercises 2 ln( 25) y C 2 ln(4 5) r C ln(6 3tan ) t C ln(2 sec ) y C ln(1 ) x C 2 ln(sec tan ) x x C 1 sec t e C csc( )t e C n( /2) ln( /6) 2[sin ] 1 v l e 2 n 0 [sin ] sin1 x l e ln(1 )r e C (1 ) ln(1 ) 1 x x x e dx x e C e 2 10 1 ln10 (log ) 2 x C 2 4 2 1 1 [ ln 2 (log ) ] 2ln 2 2 x 2 9 2 0 [ln10 (log ( 1)) ] ln10 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 8 7.2 Separable Differential Equations Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 9 differential equation Slide 4 - 10 Slide 4 - 11 Slide 4 - 12 Exercises 3 1 2 2 2 9. 3 y x C 1 2 1 3 10. 2 3 y x C 11. y x e e C 3 12. y e x C 13. 2tan y x C 3 1 2 2 3 2 14. 2 y x C
7.3I,DefinitionsandIdentitiesHyperbolic Functionslide 4-13Slide 4- 14The hyperbolic sine and hyperbolic cosine functions are defined by the equationsEvery function f that is defined on an interval centered at the origincan be written in a unique way as the sum ofone even function andsinhx-et-ecoshxteandone odd function, The decomposition is()=()+ (-)+ ()-(-)ftanhx ≥ sinhxe-ecothr =coshx+ccoshx"e+ee-ersinhx..2evenpartoddpartsechx =cschx =e+eeecoshxsinhxIf we write e* this way, we gete'+e"2-ee=2折The hyperbolic functions++bear many similaraies to theeven partoddparttrigonometricfunctions儿coshxsinhxThey describe the motions of waves in elastic solids, the shapes ofhanging electric power lines, and the temperature distributions inmetal cooling fins.Slide 4-15Slide 4- 18Th1.Exeptfor differencesinadTABLE7.4 Identitiesforsign, these resemble2lhyperbolic functionsidentitiesweknow forthetrigorniometric functionscosh° x sinb’ x = 1sinh 2r 2 sinh x coshx2. The identities are provedcosh 2x = cosh°x + sinh directly from the definitionsByetest angest:Hyperbolike sise:Bypertolc cesisecosh x - Cosh 2r + 12.sinhxcoshx=2()(t)onhstSinb° x = cosh 2r - 1llyperbelic eotaagat:r-1-5tanh’- -1 --sech x sinh 2x.coth?x=1 +cschxHyperbolic secant:Byperhodice esecantecht-ochr-1-17Slide 4-18
2016/11/15 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 13 7.3 Hyperbolic Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 4 - 14 1. Definitions and Identities Slide 4 - 15 Every function ƒ that is defined on an interval centered at the origin can be written in a unique way as the sum of one even function and one odd function. The decomposition is ( ) ( ) ( ) ( ) ( ) 2 2 f x f x f x f x f x even part odd part If we write this way, we get x e 2 2 x x x x x e e e e e even part odd part coshx sinhx They describe the motions of waves in elastic solids, the shapes of hanging electric power lines, and the temperature distributions in metal cooling fins. Slide 4 - 16 The hyperbolic functions bear many similarities to the trigonometric functions. Slide 4 - 17 Slide 4 - 18 1. Except for differences in sign, these resemble identities we know for the trigonometric functions. 2. The identities are proved directly from the definitions
Exercises Values and IdentitiesEach of Exercises I-4 gives a valuc of sinh x or cosh x. Use the defi-nitions and the identity cosh? x sinh? x 1 to find the values of theremaining five hyperbolic functions.1. sinhx = -32.shx=g417133. coshx =x>04, coshx =x>02.Derivatives and Integrals of15'3.Hyperbolic FunctionsRewrite the expressions in Exercises 510 in terms of exponentialsand simplify theresults as much as you can5, 2 cosh (ln.x)6. sinh (2 In x)7. cosh 5x + sinh 5x8, cosh 3x sinh 3x9. (sinhx + coshx)t10, In (coshx + sinh x) + In (coshx sinhx)lide 4-19Slide 4-20Exercises Finding DertvativesTABLE7.5 Derivativesof(）(一)InExrcises 1324, find the derivative ofywithrespect to the approhyperbelic functiospriate variabledad+edad13. y = 6 sinh 号14. y = sinh (2r + 1) (nh) 0ohm15. y = 2Vitunh Vi16. y = unh-oI(cobh) -ihww17, y = In(sinh z)18. y = In (eoshz2)19. y = sech e(1 In sech 9) 20. y = csch o(1 n csch o)（amha） ch层(cho- ()21 y In coh Itanh u 22 y ln sinhw Iohb u 22. coih(cobh) sch w23: y=2x, y'=223. y = (r + 1) sech (lnz)=_cosha daCHanit: Before differentiating,express in tenms of exponentials (och) -sch nhwmsinh’ μ dxand simplify.)24, y=4x,y'=424. y (4r2 ~ 1) oxch (In 2k)Icosh u dr (cschn) cschu.cothw -sinh w sinh d16.2-tamh -s8eeF！lanhf+seef13,2cosh=14. cosh(2x+1)15. V--schu cohwuoSinhz17.csh=dx=coth=18.19.sec h9-tanh.-n(seche)Lanh=sinh2cosh:20. coshe-cothe-n(csche)21. tanbySlide 4- 21Slide 4- 22EXAMPLE1TABLE7.5 Derivatives ofTABLE7.6Integral formulas for(a) (anh V1+7)= sictp VI+7-(V1+7)=sech2 V1 + hyperbolic functionshyperbolic functionsVi+t(ao) -o/-gm+c-gnmsc sinh rdi = osh w + Ccoth 5x dx =(b)(coh ) sahwcoshir dr = sinh u + C['sin='(o2-)[](c)（tanth）=Sechsech' du tanhur + C sinh2 - 1（cohm)=-chach' w dr coth w.+. C/n2102Ph4-(22-2)d(d)4e'sinhxdx l(sch) -schwahw 2sech a tanh w ahr - 0sech r + C=[e- - 2)2= (e26z- 21n2) (1 - 0) (cch) -ch rco.-csch a coth n ahi = csch w + C=42ln2-1n：1.6137Slide 4- 23Slide 4-24
2016/11/15 4 Slide 4 - 19 Exercises Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 4 - 20 2. Derivatives and Integrals of Hyperbolic Functions Slide 4 - 21 Slide 4 - 22 Exercises 13. 2cosh 3 x 14. cosh(2 1) x 1 2 15. tanh sec t h t t 1 12 16. 2 tanh sec t h t t cosh 17. coth sinh z z z sinh 18. tanh cosh z z z 19.sec tanh ln(sec ) h h 20. cosh coth ln(csc ) h 3 21. tanh v 3 22. coth v 23. 2 , ' 2 y x y 24. 4 , ' 4 y x y Slide 4 - 23 Slide 4 - 24
Evaluating IntegralsExercisesEsercisesria4Evaluate the integrals in Exercises 4160,52.tanh 2r dt31coth.xdr42.41.sinh 2x dtsinhd53.20'cosh e d54,4esinhed)43. 6coh(-1n3)dr444 0osh (3x In 2) dr2 sinh (sin e) cos e d55.cosh (tan f) sec° e de5645;/tanhdx46.dcothcosh (In )so ViaV57.58,Vr47. / sap ( ~ )ad48.esch (5 x) dxs ()at4sin(6)60.59,csch (ln t) coth (In t) atsech Vf tanh Vr ar5052.1749.53.号+1n25.054.2hn2-3Vi283243. sinke号-ln3)+C42.5cosh=+C41.0sh2x+C57.355. e-!56. e+258. 8e+!e-e)-46.JEln(sinh45: 71n(cosh )+CJse44.sinb(3xIn2)+C99359.+1m2-2in1060821049:-2sechf+C47.tanh(r-)+(S0..-csehinn+C48,coth(5-x)+Cslide 4-25Slide 4- 283.InverseHyperbolicFunctions56The inverses ofthe sixbasic hyperbolicFIGURE7,5ThegraphsofntxNotsymmetries about thefunctions areveryusefulin integrationliney-.side 4- 27Slide 4- 28sinh+= In(r+ V2 + 1),00:00cosh-x.= In(x + V-1).x≥ 1474μ/<11+V-sech' x.= In0<x≤1Vi+xcsch"r - In (I +++0[x|couh-]三>！GURETpsothe iyolictng,coangt,andcSlide 4- 29Slide 4-30
2016/11/15 5 Slide 4 - 25 Exercises 1 41. cosh 2 2 x C 42. 5cosh 5 x C 43. sinh( ln 3) 2 x C 4 44. sinh(3 ln 2) 3 x C 45. 7ln(cosh ) 7 x C 46. 3 ln(sinh ) 3 C 1 47. tanh( ) 2 x C 48. coth(5 ) x C 49. 2sec h t C 50. csc (ln ) h t C Slide 4 - 26 Exercises 5 51. ln 4 1 17 52. ln 2 8 3 53. ln 2 32 3 54. 2ln 2 4 1 55. e e 1 56. 2 e e 3 57. 4 2 2 1 58. 8( ) e e e e 3 1 59. ln 2 8 2 99 60. 2ln10 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 4 - 27 3. Inverse Hyperbolic Functions The inverses of the six basic hyperbolic functions are very useful in integration. Slide 4 - 28 Slide 4 - 29 Slide 4 - 30