Integration by Substitution: A Case Study in the Relationship Between Calculus and Analysis

SummaryDo the theorems we prove in our analysis classes justify the calculations we teach in calculus classes? In this article I use some examples of integration by substitution to show that the answer is more complicated than one might think. Notes1 Courant [2, pp. 211–212] gives a different justification for this kind of substitution in definite integrals, using limits of Riemann sums. Surprisingly, he assumes that g has a continuous, nonvanishing derivative, even though his proof uses only (g−1)′ and not g′.Additional informationNotes on contributorsDaniel J. Velleman Daniel J. Velleman (djvelleman@amherst.edu) received his B.A. from Dartmouth College in 1976 and his Ph.D. from the University of Wisconsin–Madison in 1980. He taught at the University of Texas before joining the faculty of Amherst College in 1983. Since 2011, he has also been an adjunct professor at the University of Vermont. He was the editor of the American Mathematical Monthly from 2007 to 2011. In his spare time he enjoys singing, bicycling, and playing volleyball.