Consider the integral a

MATHEMATICS · Multi-part · AP · Calculus BC · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Consider the integral a.

I=\displaystyle \int x\,\ln x\,dx

. Two students propose different initial decisions about which factor will be differentiated and which will be integrated when evaluating

I

. • Student A plans to differentiate

x

and integrate

\ln x

. • Student B plans to differentiate

\ln x

and integrate

x

. Respond to the prompts below, providing clear reasoning rather than a computational solution. (a) Identify which student’s decision is more efficient for evaluating

I

. Explain your choice by comparing the relative difficulty of finding derivatives versus antiderivatives for the two factors. (b) Predict whether the more efficient decision will require more than one iteration of the chosen technique to complete the evaluation of

I

. Justify your prediction qualitatively without performing any algebraic integration.

Related theory

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