Laplace Transform of $\cosh(at)/(at)$

Question

Can someone give me a clue on how to compute this Laplace transform?

$$\mathcal{L}\left[ \frac{\cosh(at)}{at} \right]$$

Answer 1

One of the properties of the Laplace transform is that

$$\mathcal{L} \left( \frac{f(t)}{t} \right) = \int_s^{+ \infty} F(s) \, ds.$$

This means that

$$\mathcal{L} \left( \frac{\cosh(at)}{at} \right) = \frac{1}{a} \int_s^{+ \infty} \mathcal{L}(\cosh(at)) \, ds.$$

Take from here.

Answer 2

By definition,$$\mathcal{L}\left\{\frac{\cosh(at)}{at}\right\}=\int_0^\infty e^{-st}\frac{\cosh(at)}{at}dt \\ = \frac{1}{2a}\int_0^\infty e^{-st}\frac{e^{at}+e^{-at}}{t}dt \\ = \frac{1}{2a}\int_0^\infty \frac{e^{t(a-s)}+e^{-t(a+s)}}{t}dt \\=\frac{1}{2a} \int_0^\infty \frac{e^{t(a-s)}}{t}dt+ \frac{1}{2a} \int_0^\infty \frac{e^{-t(a+s)}}{t}dt$$ Also, the Laplace transform is not defined for some $a \in \Bbb{R}$; you should find those values before integrating.