Partial Fractions, and Integration by Partsrestart;In this worksheet, we show how to explicitly implement integration by parts, and how to convert a proper or improper rational fraction to an expression with partial fractions.Partial FractionsWe consider the function NiMvLSUiZkc2IyUieEcqJiwuKiYiIiMiIiIqJClGJyIiJkYsRixGLComIiIpRiwqJClGJyIiJUYsRiwhIiIqJiIjOkYsKiQpRiciIiRGLEYsRiwqJiIjNUYsKiQpRidGK0YsRixGNSomIiIqRixGJ0YsRjUiI0ZGLEYsLCxGMkYsKiZGNEYsRjhGLEY1KiZGL0YsRj1GLEYsKiZGNEYsRidGLEY1RjRGLEY1.f:=x -> (2*x^5-8*x^4+15*x^3-10*x^2-9*x+27)/(x^4-4*x^3+5*x^2-4*x+4);We use Maple to find its integral.Int(f(x),x)=int(f(x),x);We now wish to use Maple to convert the integrand above by the method of partial fractions. When doing partial fractions by hand, we use the method only with proper fractions. Here, we use the convert command, where the argument parfrac refers to partial fraction format. This will work with improper fractions also. pf:=convert(f(x),parfrac,x);Int(pf,x)=int(pf,x);This is the same answer that we got above. To illustrate further, let us create separate expressions for the numerator and denominator of NiMtJSJmRzYjJSJ4Rw==. We will use the numer and denom commands to do this.numerator:=numer(f(x));denominator:=denom(f(x));We form the original improper rational fraction.original:=numerator/denominator;Since we have an improper rational fraction and the method of partial fractions is for proper rational fractions (degree of numerator less than degree of denominator), we use quo and rem to get the quotient and remainder from a long division of polynomials. Notice that the arguments for each command are "dividend, divisor, variable."quotient:=quo(numerator,denominator,x);remainder:=rem(numerator,denominator,x);We form a rational fraction by dividing the remainder by the divsor.rf:=remainder/denominator;We rewrite the original improper fraction as the sum of the quotient and a proper fraction.new:=quotient+rf;Again, we can use the convert command to convert the proper rational expression to partial fractions.rf:=convert(rf,parfrac,x);The entire integrand is the sum of the quotient and the partial fraction decomposition of the proper fraction.integrand:=quotient+rf;Integration by PartsMaple has a student package which is designed to illustrate calculus concepts in a step by step manner. We load this package by using the with statement.with(student);A list is given of all the new commands added. Our interest is in intparts. Let us apply this to NiMtJSRpbnRHNiQqJiUieEciIiItJSRleHBHNiMqJiIiJkYoRidGKEYoRic=. The intparts command takes two arguments. The first in the inert intergal we are interested in, and the second is the NiMlInVH from NiMtJSRpbnRHNiQlInVHJSJ2Rw==.Int(x*exp(5*x),x)=intparts(Int(x*exp(5*x),x),x);The command intparts can also be used with definite integration.Int(x*exp(5*x),x=2..4)=intparts(Int(x*exp(5*x),x=2..4),x);Change of VariableWe can also use changevar for a change of variable. For example, suppose we wish to use the substitution NiMvJSJ1RyomIiIjIiIiJSJ4R0Yn in the integral NiMtJSRpbnRHNiQqJiIiIkYnLSUlc3FydEc2IywmRidGJyomIiIlRicqJCklInhHIiIjRidGJyEiIkYyL0YwOyUiYUclImJH.Int(1/sqrt(1-4*x^2),x=a..b)=changevar(u=2*x,Int(1/sqrt(1-4*x^2),x=a..b),u);