Mathematics

Mathematics

Problem

For each improper integral below, decide whether it converges or diverges. Give a clear, concise justification for every decision. Integral I: ∫0∞e−2x dx\displaystyle \int_{0}^{\infty} e^{-2x}\,dx Integral II: ∫1∞1x dx\displaystyle \int_{1}^{\infty} \frac{1}{x}\,dx Integral III: ∫0∞xe−x2 dx\displaystyle \int_{0}^{\infty} x e^{-x^{2}}\,dx Integral IV: ∫1∞1x2/3 dx\displaystyle \int_{1}^{\infty} \frac{1}{x^{2/3}}\,dx (a) Evaluate Integral I or show that it diverges. (b) Determine whether Integral II converges or diverges and justify your answer. (c) Determine whether Integral III converges or diverges and justify your answer. (d) Determine whether Integral IV converges or diverges and justify your answer.

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