Modified Bessel function of the second kind

Questions
1

The modified Bessel functions of the second kind K_m can be represented by:

K_m.png

I tried to calculate this integral with Mathematica Version 12:

f[t_] := Exp[-x*Cosh[t]]*Cosh[m*t];
Integrate[f[t], {t, 0, Infinity}]

Unfortunately Mathematica does give no result by solving the integral above. Can someone explain me, how to solve this integral with Mathematica.
Thank you for your help in advance.
Ruediger

2

Check out the inbuilt function BesselK

3

Thank you, but I know this inbuilt function BesselK. My hope was that Methematica will give me This answer by solving the integral above. Mathematica writes only the integral as result.