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Yeah it would be useful if someone explains what a positive integral solution is.
Actually I did a research abou this myself and I found out in a physical chemistry book that logarithmic equations such as the above are called "transcendental equations", equations that do not have a solution in a closed form. The solutions are rather found using a mathematical software. I had that in mind although I thought there could be some sort of a mathematical trick or something that can be applied.
Oren
I = ∫ [ 1 / ( 1 + cosx ) ( 1 + cosx) ] dx
t = 1 / 1 + cosx
dt = [ sinx / ( 1 + cosx)( 1 + cosx) ] dx
Cosx = 1 / t - 1 so Sinx = sqrt [( 2t - 1 / t )] / t
I = ∫ [ t / sqrt( 2t -1 ) ] dt = 1/2 ∫ [ ( 2t +1 - 1 ) / sqrt(2t - 1 ) ] dt
I = 1 / 2 ∫ [ sqrt ( 2t -1 ) - 1 / sqrt ( 2t - 1 ) ] dt
I = 1 /2 sqrt(2t -1) [ ( 2t -1 ) / 3 - 1 ]
If im not mistaken the final expression should be something like this :
I = 1 /2 sqrt ( 1 - Cosx / 1 + Cosx ) [ (1 /3) (1 - Cosx) / ( 1 + Cosx ) - 1 ] + C
The definite integral can then be evaluated. Sorry for this unclearness as I still havent learned about LAtex
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