Consider the rational function R(x)= 3x^ 2 +4x+7 x^ 2 +2x+5
MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
Consider the rational function
R(x)=x2+2x+53x2+4x+7.
A student claims that R(x) can be rewritten as a sum of two simpler functions whose antiderivatives are familiar from the standard AP Calculus formulas.
(a) Show one algebraic pathway to decompose R(x) into the sum of a polynomial term and a proper rational term whose numerator is linear in x. Clearly display each intermediate expression.
(b) Rewrite the denominator of the remaining proper fraction so that its structure matches a known basic antiderivative form. Justify algebraically that your rewritten form is equivalent to the original denominator.
(c) State (without performing any integration) the types of elementary functions that will appear in the antiderivative of R(x). Provide a brief justification for each type you list.