2 x x 2 a 2 2 3 8 a 4 x x 2 a 2 3 8 a 5 arctan x a C Problems 22 a Compute the

# 2 x x 2 a 2 2 3 8 a 4 x x 2 a 2 3 8 a 5 arctan x a c

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2·x(x2+a2)2+38a4·xx2+a2+38a5arctan(xa)+C.Problems 2.2.(a) Compute the integralI=x2+3(2x5)3dx;(b) Compute the integralI=dxa2sin2x+b2cos2x;(c) Compute the integral3x(lnx)2dx;(d) Compute the integral3xcosxdx;(e) Derive the reduction formula for the integration ofIn=dxsinnxand use it for calculating the integralI3=dxsin3x.Solution.Problem??(a): We apply the substitutiont= 2x5, i.e.x=t+52anddx=dt2, to getI=x2+ 3(2x5)3dx=18t2+ 10t+ 37t32dt=18(t12+ 10t12+ 37t32)dt=14[t2+ 30t1113t12]+C=x2+ 10x5932x5+C.Problem??(b): We apply the substitutiont= tan(x), i.e.dt=dxcos2x, as followsI=dxa2sin2x+b2cos2x=1a2tan2x+b2·dxcos2x=dta2t2+b2=1a2dtt2+(ba)2=1abarctan(abtanx)+C.11
Problem??(c): We apply the integration by parts method:3x(lnx)2dx=34x43(lnx)232x13lnxdx=34x43(lnx)232[34x43lnx34x13dx]=x43(34(lnx)298lnx+2732)+C=332x43(8(lnx)212 lnx+ 9)+C.Problem??(d): We can use the formula from Example??:3xcosxdx=exln 3cosx dx=14exln 3(sinx+ lnxcosx) +C.Problem??(e): Assumen2 and apply the Integration by Parts Rule:In=dxsinnx=sin2x+ cos2xsinnxdx=In2+coscosxsinnxdx=In21n1cosxsinn1x1n1In2=n2n1In2cosx(n1) sinn1x.Notice that by applying the substitutiont= cosxwe getI1=dxsinx=sinx dx1cos2x=dt1t2=12∫ (1t11t+ 1)dt=12ln1t1 +t+C=12ln(1cosx)2sin2x+C= ln1cosxsinx+C= ln|cscxcotx|+C.Consequently, by the reduction formulaI3=12I1cosx2 sin2x=12ln|cscxcotx| −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 indefinite12
integrals can be expressed by elementary functions, and that means that theoretically, as long as we areable to effectuate related algebraic computations (like computations of roots of the denominator), wewill also be able to find an exact formula for the integral. Moreover, many other types of integrals canbe 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:ex2dx, ,sinx2dx,cos2xdxsinxxdx,cosxxdx,dxlnx,but there are many more.