A student attempts to find an antiderivative for the integrand 3x^ 2 x^ 3 +1 and…
MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
A student attempts to find an antiderivative for the integrand
x3+13x2
and writes the following in her notebook:
"Since the numerator is the derivative of the denominator, an antiderivative is
G(x)=(x3+1)2+C."
Assume no arithmetic mistakes are present in the student’s differentiation steps—focus only on conceptual or algebraic reasoning.
(a) Differentiate the student’s proposed G(x) and simplify the result completely.
(b) Compare your result from part (a) with the original integrand. Identify the specific mismatch and explain why the student’s reasoning is incorrect.
(c) Construct a correct antiderivative G(x) for x3+13x2 and verify your answer by differentiation.
(d) Generalize: For any differentiable function u(x) with u′(x)=0, state a condition under which u′(x)u(x)n is the derivative of n+1u(x)n+1, and briefly justify your statement.