IntCalc.pdf - Chapter 5 Integral Calculus 5.1 Introduction...

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Chapter 5Integral Calculus5.1IntroductionAlmost all of you would have seen several integration methods before arriving at UCL and theElementary Techniques Test has been checking your knowledge of simple examples of integrationby substitution, by parts and by the simplest cases of partial fractions. In this part of the modulewe will revisit these approaches and look at some more complicated cases.A(x) =Zxafx) d˜x.A(x+h) =Zx+hafx) d˜x=Zxafx) d˜x+Zx+hxfx) d˜xA(x) +hf(x).But alsoA(x+h) =A(x) +hA0(x) +· · ·,sof(x) =A0(x). This is the fundamental theorem of calculus:f(x) =ddxZxafx)d˜x.5.2Hyperbolic functionsHyperbolic functions arise in many areas. Our motivation for introducing them now is that theyare good functions to use in the evaluation of certain integrals.Recall that the ordinary trigonometric functions can be defined in terms of complex expo-nentials bycosx=12(eix+ e-ix),sinx=12i(eix-e-ix).The hyperbolic functions are the “real analogues” of these definitions:coshx=12(ex+ e-x),sinhx=12(ex-e-x).These are usually called the hyperbolic cosine and hyperbolic sine functions respectively, whenspeaking we usually abbreviate these to “cosh” and “shine”, although in North America sinhis often pronounced “sinch”. Notice that the “h” in “cosh” is part of the name, and not someconstant appearing in the argument!Recall that a functionfis calledeveniff(-x) =f(x) for allxin its domain andoddiff(-x) =-x. The functions cos and cosh are both even and the functions sin and sinh are both33
CHAPTER 5.INTEGRAL CALCULUS34odd. Most functions are neither even nor odd but every function defined on the real line (or anydomain symmetric about 0) can be written uniquely as the sum of an even and an odd function,called the even and odd parts of the function. With this language we can say that cos and sinare the even and odd parts of the function eixsinceeix= cosx+ i sinxand cosh and sinh are the even and odd parts of the function exsinceex= coshx+ sinhx.Just like their usual trigonometric counterparts, the hyperbolic trigonometric functions satisfymany identities. In particular(coshx)2= 1 + (sinhx)2.This identity explains the origin of the term “hyperbolic”.The parametrisationx= cost,y= sintis a parametrisation of the circlex2+y2= 1 because (cost)2+ (sint)2= 1.Forthis reason these “ordinary” trigonometric functions are sometimes referred to as “circular”trigonometric function.The parametrisationx= cost,y= sinhttraces out the right side(x >0) of the hyperbolax2-y2= 1 because (cosht)2-(sinht)2= 1.It also follows immediately from the definition thatd sinhxdx= coshxandd coshxdx= sinhx.Letting tanhx= sinhx/coshx, sechx= 1/coshxetc., we also haved tanhxdx= (sechx)2.