How would I integrate this?

Hi,

I'm after integrating the function 1 / (e^x + e^-x); how exactly would I go about this? Some kind of substitution seems like the best route, but I'm not entirely sure what.

Thanks,
Stewart

Welcome to the forums.

I believe this is 1/2 hyperbolic secant, or sech(x)/2. Aren't hyperbolic trig functions analagous to standard trig functions? In other words, I think you'd use the same method that would be used to integrate sec(x)/2. Just my recollection from a long time ago......:)

Also, isn't this the same as

(e^x + e^-x)^(-1)

I think you can simplify this a lot more:

1/ e^x + e^-x

=

1/ (e^x + 1/e^x)

= 1/ (e^x^2 + 1 / e^x)

= e^x/ e^x^2 + 1

u = e^x
du = e^x (cancels with e^x on top)

= 1/ (u^2+1)

= arctan(e^x)

Looks like (1/a)arctan(x/a) to me, but I could have easily messed up somewhere (this is tough without paper and pencil).

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