
Re: History of the integral of secant
Posted:
Feb 28, 2009 9:54 PM


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)ln1  sin x + (1/2)ln1 + 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
= lnsec x + tan x + C
Regards,
Rick
<snip>

