The integral of 1/x

摘要

Students meeting the result ∫1/x-dx=1nx+C for the first time are often amazed by the fact that a function so unlike x~(n+1) can 'fill the gap' at n = -1 in the integration of x~n. Here we show that In x must fill that gap,by examination of the limit of | x~n dx as n→-1. Consider the functiondefined byI(n,b)=∫_1~b t~n dt,n≠-1where b is a positive constant.