Using double integrals to solve single integrals. Issue 548 (14th June 2016)

• Record Type: Journal Article
• Authors: Jameson, G. J. O.
• Is Part Of: Mathematical gazette. Volume 100:Issue 548(2016)
• Page Start: 257
• Page End: 265
• Publication Date: 2016-06-14
• Subjects: Mathematics -- Periodicals; Mathématique; Mathematik; Mathematics; Periodicals; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); Zeitschrift; Online-Publikation; 510
• DOI: 10.1017/mag.2016.62
• Languages: English
• ISSNs: 0025-5572

Abstract

Abstract : Consider the integral where b > a > 0. First, let us clarify why it even exists. Of course, convergence at infinity is ensured by the exponential terms, but the integrals of and e – ax / x and e – bx / x, taken separately, are divergent at 0, since these integrands equate asymptotically to 1/ x as x → 0. However, so ( e – ax – e – bx )/ x tends to the finite limit b – a as x → 0 and there is no problem integrating it on intervals of the form [0, r ]. A neat way to evaluate I 1 starts by expressing the integrand itself as an integral: (1) Inserting this into I 1 converts it into a double integral.