Convergence of Bessel integral function

Question

I have no idea how to prove the convergence. I am told I have to use the comparison test. I have split the integral from $0$ to $1$ and from $1$ to $+\infty$. I know $\cosh(t)\geqslant 1$ and $e^{-x}>0$. But now I am stuck.

Answer 1

$$K_\alpha(x) \approx \frac{1}{2}\int_{0}^{+\infty}\exp\left(-\frac{x}{2}e^t+\alpha t\right)\,dt = \frac{1}{2}\int_{1}^{+\infty} z^{\alpha-1}e^{-xz/2}\,dz$$ and: $$ \int_{0}^{+\infty} z^{\alpha-1}e^{-xz/2}\,dz = \left(\frac{2}{x}\right)^{\alpha}\Gamma(\alpha).$$