43²8(lnx)2°12 lnx+ 9³+C:Solution:For(d): Since3x=exln 3We can use the formulaZ3xcosxdx=Zexln 3cosx dx=14exln 3(sinx+ lnxcosx) +C:Solution:For(d): Assumen³2and apply the Integration by Parts Rule:In=Zdxsinnx=Zsin2x+ cos2xsinnxdx=In°2+Zcoscosxsinnxdx=In°2°1n°1cosxsinn°1x°1n°1In°2=n°2n°1In°2°cosx(n°1) sinn°1x:Notice that by applying the substitutiont= cosxwe getI1=Zdxsinx=°Z°sinx dx1°cos2x=°Zdt1°t2=12Z °1t°1°1t+ 1±dt=12ln¸¸¸¸1°t1 +t¸¸¸¸+C=12ln¸¸¸¸(1°cosx)2sin2x¸¸¸¸+C= ln¸¸¸¸1°cosxsinx¸¸¸¸+C= lnjcscx°cotxj+C:Consequently, by the reduction formulaI3=12I1°cosx2 sin2x=12lnjcscx°cotxj °cosx2 sin2x+C:3Integration of Rational FunctionsAmong the elementary functions, the rational functions played a special role in the development ofintegration techniques. One of the important features of rational functions is the fact that their inde°niteintegrals can be expressed by elementary functions, and that means that theoretically, as long as we areable to e/ectuate related algebraic computations (like computations of roots of the denominator), wewill also be able to °nd an exact formula for the integral. Moreover, many other types of integrals can11
be reduced, by special substitutions, to integrals of rational functions. However, this is not the case forall elementary functions. There are many known functions with integrals that can not be expressed byelementary functions, i.e. it is impossible to express them by an analytic formula using basic elementaryfunctions, arithmetic operations, their compositions and inverses. The following are just few well knownintegrals that can not be expressed by elementary functions:Ze°x2dx; ;Zsinx2dx;Zcos2xdxZsinxxdx;Zcosxxdx;Zdxlnx;but there are many more.A rational functionP(x)Q(x), whereP(x)andQ(x)are two polynomials, is called aproper rational functionifdegP(x)<degQ(x):By using thedivision algorithmfor polynomials, every rational functionf(x) =P1(x)Q(x)can be representedas a sum of a polynomial and a proper rational function, i.e.f(x) =P1(x)Q(x)=M(x) +P(x)Q(x);whereP1(x) =M(x)Q(x) +P(x);withP(x)being the remainder of the division ofP1(x)byQ(x), i.e.degP(x)<degQ(x). We say that aproper rational functionP(x)Q(x)isirreducibleifP(x)andQ(x)have no common linear or quadratic factors.Lemma 3.1Suppose thatP(x)Q(x)is a proper rational function such thatQ(x) = (x°²)kQo(x), where²is not a root ofQo(x)andk2N. Then there exist a constantAand a polynomialPo(x)such thatP(x)Q(x)=A(x°²)k+Po(x)(x°²)k°1Qo(x);wheredegPo(x)<degQ(x)°1.