On Mar 1, 6:38 pm, "Stephen J. Herschkorn" <sjhersc...@netscape.net> 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. > > -- > Stephen J. Herschkorn sjhersc...@netscape.net > Math Tutor on the Internet and in Central New Jersey
I agree that the approach taken in most Calculus texts is unmotivated. The following approach strikes me as more motivated although probably equally ahistorical. Pretend that you really don't know the antiderivative of sec x but know the other basic facts a Calc II student should know. How might you discover the formula? Well - since it is an antiderivative you are looking for, ask yourself if you know of any function which has derivative sec x. The answer is of course no - so ask yourself if you know of any function whose derivative *involves* sec x. The answer is yes - including two basic ones:
(sec x)' = sec x tan x (tan x)' = sec^2 x
How does this help? Can these be combined in some way to get an antiderivative? It seems like a natural attempt to see what happens when you add these two equations together, if for no other reason than that is often a common move in mathematics. This leads to
(sec x)' + (tan x)' = sec x tan x + sec^2 x = (sec x)(tan x + sec x)
(sec x + tan x)' = (sec x)(sec x + tan x)
(sec x + tan x)'/(sec x + tan x) = sec x
and a bright enough student should be able to recognize the LHS as (ln| sec x + tan x|)'.