# Integration by long division

Since integration is the inverse of differentiation, you can think of integration as “dividing” by d.

J. P. Ballantine [1] shows that you can formally divide by d and get the correct integral. For example, he arrives at

using long division!

[1] J. P. Ballantine. Integration by Long Division. The American Mathematical Monthly, Vol. 58, No. 2 (Feb., 1951), pp. 104-105

## 3 thoughts on “Integration by long division”

1. Bill Meisel

John – isn’t this just tabular integration by parts written in a slightly different form?

2. Rick Bryan

This is reminiscent in its way of Oliver Heaviside’s Operator Calculus, where he blithely multiplied and divided his operators — even raising them to fractional powers — and happily manipulated divergent series, all while revolutionizing how electrical systems were analyzed. Personally he showed many signs of being a crank, with remarkable disdain for those who carefully explained why “you can’t just do that.” Although he enjoyed remarkable success (for his time) I’m glad I was able to use Laplace transforms when I was an E.E. undergrad.