In article <49AB1C92.firstname.lastname@example.org>, "Stephen J. Herschkorn" <email@example.com> wrote:
> Stephen J. Herschkorn wrote: > > > What is the history of the trick to find the indefinite integral of > > the secant function? Who first used it, and when? > > > Thanks to all for the responses to my query. From these, I summarize > that Mercator (whose work precedes that of Newton and Leibniz by about > two centuries) showed that integral(x=0..t, sec x) = ln |tan t/2 + > pi/4|, though not in that notation. I suspect that, after the > development of calculus, if one notices that ln |tan t/2 + pi/4| = ln > |sec t + tan t|, one can then come up a posteriori with the trick of > converting sec x to sec x (sec x + tan x) / (sec x + tan x ). > > As I noted in another post, this all comes up because I was presenting > the integral of secant, via the aforementioned trick, in a > second-semseter calculus class the other day. (I have been teaching > calculus for only two years now, and this is the first time this topic > has come up in the syllabus.) The students very naturally asked, "How > did you know to multiply by sec x + tan x?" My response was, well, > that's just a trick someone came up with. I wouldn't expect the > students to come up with it on their own. > > Rick Decker, in another post in this thread, posted another derivation. > Rewriting it a bit, > > sec x = 1 / cos x = cos x / cos^2 x = cos x / (1 - sin^2 x). > > Now use the substitution u = sin x and apply partial fractions to get > > 1/2 [ln(1 - sin x) + ln(1 + sin x)] > > as an antiderivative. Now use the properties of logarithms and the > trigonometric functions to show that this equals ln |sec x + tan x|. > > This derivation seems more "natural" to me: Asked to find the > antiderivative of the secant, I see nothing unusual in coming up with > it. Once one sees the final result, one can then come up with the > efficient trick. I wouldn't be surprised if this how it all went down.
Find the antiderivative of sec(x) with the z = tan(x/2) substitution. It is widely useful, and therefore perfectly natural to try.