To establish convergence of the improper integral

MATHEMATICS · Single correct · A Level · Mathematics · Integration · Integration

Problem

To establish convergence of the improper integral ∫1∞e−x ln⁡x dx\displaystyle \int_{1}^{\infty} e^{-x}\,\ln x\,dx, a student plans to differentiate one factor of the integrand and integrate the other. Which choice of the factor to be differentiated most directly produces an integrand whose magnitude can be bounded by a simple exponential function? (a) Differentiate e−xe^{-x} (b) Differentiate e−x ln⁡xe^{-x}\,\ln x as a single unit (c) Differentiate ln⁡x\ln x (d) Differentiate xx

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