MATHEMATICS · Multi-part · AP · Calculus BC · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
Evaluate the improper integral
L=∫1∞x2lnxdx.
Your response should address the following prompts.
(a) State a strategic choice of which factor to differentiate and which to integrate that will lead to a convergent expression when the upper limit approaches ∞. Justify your choice by analyzing the asymptotic behavior of each factor after one iteration.
(b) Carry out the complete evaluation of L, including all limit processes needed to confirm convergence.
(c) Suppose the opposite initial decision were made. Describe how that decision would affect the convergence of intermediate expressions and the overall tractability of the problem.