Hyperbolic integral

[royal plover]

Stuck in in my integral

[harsh lark]

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[frail cosmos]

Let's see.
We have √(cosh(2x))dx. We take u = sinh(2x).
du = 2cosh(2x)dx
cosh(2x) = √(1 + u^2)
So:
√(cosh(2x))dx = (1/2)(2cosh(2x))dx/√(cosh(2x)) = (1/2)du/(1 + u^2)^(1/4)
So yeah, that's fine. And by Chebyshev's substitution criteria this can be shown to be non-elementary.

Some other trickier approach is needed, probably.

[royal plover]

Thanks for verifying my derivation!

can we conclude then that an elementary primitive can't be found?

[frail cosmos]

Yeah.

Wonder how we can find the integral, though.

[kind sun]

I am sorry to inform you that it doesn't converge

the integrand is lower-bounded by 1 trivially

[royal plover]

Been asking the same question. I would be surprised if the only way to find a solution is through elliptic integrals

[kind sun]

no

[frail cosmos]

Oh. Duh 😅

[kind sun]

sqrt(cosh² + sinh²) >= sqrt(1) = 1

[frail cosmos]

I kept thinking about 1/√(cosh(2x)) for some reason...

[royal plover]

so the whole think is wrong?

[kind sun]

well no it's not wrong, all your algebraic manipulations are correct

it's just useless

because the integral doesn't converge

[frail cosmos]

Yeah...

[royal plover]

Hmm ff.
That's a bit annoying since I was trying to resolve this integral as the arc length of the parametric curve y^2 - x^2 = 1

parametrised by (sinh(t), cosh(t))

[kind sun]

well it's clearly infinite

[frail cosmos]

Yeah, it's a hyperbola, after all.

[royal plover]

yeah I overlooked that

[kind sun]

it's the two curves y = sqrt(1+x²) and y = -sqrt(1+x²)

[royal plover]

thanks for pointing it out! would have kept thinking about it lol

[kind sun]

no worries, it happens sometimes

[royal plover]

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