Well, as I see it, the definition of antiderivative of a function g is a function f such that f is differentiable and f'(x) = g(x) for all x in the domain of f. Now, 1/x is defined on R - {0}. And on that set, the most general antiderivative of 1/x is ln( x) + C_1, x > 0 f(x) = ln(-x) + C_2, x < 0 --Dan On 2014-02-04, at 7:11 PM, Eugene Salamin wrote:
As long as you don't cross the singularity at 0, G. B. Thomas is correct. Assume b > a > 0.
int(1/x, x=-b..-a) = -int(1/x, x=a..b) = -log(b/a).
log|-a| - log|-b| = log a - log b = -log(b/a).
If you do cross the singularity, then in integral has no well defined limit.
I wrote: ----- One error I believed for many years* is that the antiderivative of 1/x is ln|x| + C. --Dan ______________________________________________ * Thanks to my misplaced trust in G.B. Thomas. -----