# Integral Form of Modified Bessel Function of the Second Kind

Why can't Mathematica integrate

r = Integrate[Exp[-x Cosh[t]], {t, 0, Infinity}];
r = Assuming[Element[x, Reals], Simplify[r]];
Together[r]


From Wikipedia, it should be $K_0(x)$, the modified Bessel function of the second kind for $\alpha =0$, since$$K_\alpha(x) = \int_0^\infty \exp(-x\cosh t) \cosh(\alpha t) \,dt.$$ And there is definitely a BesselK function in Mathematica. It seems like a similar problem to here.

• It has been like that for a while; I'm not sure why. In any case, try making the substitution $u=\cosh t$ before using Integrate[]. – J. M. is away Feb 27 '16 at 4:23
• To elaborate on the comment by @J.M., Integrate[Exp[-x u]/Sqrt[u^2 - 1], {u, 1, Infinity}, Assumptions -> x > 0] gives the desired result. – bbgodfrey Feb 27 '16 at 5:00
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