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Topic: Historyu of the integral of secant
Replies: 13   Last Post: Mar 3, 2009 3:11 PM

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Michael Press

Posts: 2,137
Registered: 12/26/06
Re: History of the integral of secant
Posted: Mar 3, 2009 3:11 PM
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In article <49AB1C92.7050209@netscape.net>,
"Stephen J. Herschkorn" <sjherschko@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.


Find the antiderivative of sec(x) with the z = tan(x/2)
substitution. It is widely useful, and therefore perfectly
natural to try.

--
Michael Press



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