Two rational functions are defined by R1(x)=x+23x2+4x+1,R2(x)=x2+6x+9x2+6x+10. For each function, an antiderivative is to be found analytically.
(1) Part 1: For R1(x), state an algebraic transformation that must be carried out before antidifferentiation can begin, and justify your choice by referring explicitly to the degrees of the numerator and denominator.
(2) Part 2: For R2(x), identify the algebraic manipulation that would most directly prepare the integrand for antidifferentiation, and justify your choice by analyzing the structure of the denominator.
(3) Part 3: Compare the two situations: explain why the transformations you proposed in parts (a) and (b) differ even though both functions are rational. Discuss the implications each transformation has for the form of the antiderivative.