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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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Rick Decker

Posts: 1,356
Registered: 12/6/04
Re: History of the integral of secant
Posted: Feb 28, 2009 9:54 PM
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On Feb 28, 3:26 pm, "Stephen J. Herschkorn" <sjhersc...@netscape.net>
wrote:
> Michael Press wrote:
> >In article <49A8C201.20...@netscape.net>,
> > "Stephen J. Herschkorn" <sjhersc...@netscape.net> wrote:


<snip>
>
> I refer to the trick of noting that sec x = sec x (sec x + tan x) / (sec
> x + tan x). (I was teaching this the other day and pointed out that I
> would never expect the students to come up with this on their own.) I
> see what you are saying - people noticed a numerical identification in a
> different context, nearly two centuries before Newton and Leibniz. So
> mathematicians had a specific formula and could look for ways to derive
> it. I still wonder when this trick was first known. For example, does
> it go back to Newton or Leibniz?


I don't know, but there is a solution which might predate the trick.
Omitting the differential for simplicity:

Int[sec x] = Int[1 / cos x] = Int[cos x / (cos x)^2]

= Int[cos x / (1 - (sin x)^2]

= Int[(cos x)(1 / ((1 - sin x)(1 + sin x)))]

= Int[((cos x) / 2)(1 / (1 - sinx) + 1 / (1 + sin x))]

= (1/2)Int[(cos x) / (1 - sin x)] + (1/2)Int[(cos x) / (1 + sin x)]

= (-1/2)ln|1 - sin x| + (1/2)ln|1 + sin x| + C

= (1/2)ln|(1 + sin x) / (1 - sin x)| + C

= (1/2)ln|(1 + sin x)^2 / (1 - (sin x)^2)| + C

= (1/2)ln|(1 + sin x)^2 / (cos x)^2| + C

= ln|(1 + sin x) / cos x| + C

= ln|sec x + tan x| + C


Regards,

Rick

<snip>



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