Let $B$ be a standard Brownian motion, and consider the drift process $X_t=B_t+ct$ for $c\in \mathbb R$. For $x>0$, set $H_x=\inf\{t>0: X_t=x\}$. I have been able to show that this is a stopping time, and I have been able to calculate its Laplace transform $$\mathbb E[e^{-\lambda H_x}]=e^{-x(\sqrt{c^2+2\lambda}-c)}.$$
Now, I want to calculate $\mathbb P\{H_x<\infty\}$ using the Laplace transform, somehow. I confess that this is part of an exercise in a lecturer's notes, and the way the question is asked implies there is an easy way to calculate $\mathbb P\{H_x<\infty\}$ using $\mathbb E[e^{-\lambda H_x}]$. However nowhere in the exercise is the name Laplace mentioned, and in the notes I cannot find anything relating the Laplace transform (named or not) of an r.v to its distribution. Nonetheless, I figured this must be the approach the exercise was implying. Specifically that there was some link between the Laplace transform and distribution function of an r.v. So after some Googling I came across these notes, which state that for a non-negative r.v $X$ we have: $$F_X=\mathcal L^{-1}_\lambda\left (\frac{\mathbb E[e^{-\lambda X}]}{\lambda}\right).$$ These notes do not justify why this is the case, so I would appreciate a reference to a source detailing this relationship (I have a feeling it involves the characteristic function, but my knowledge is too sketchy at the moment to make this link concrete for myself).
I have little experience with the Laplace transform, so after some further Googling I came to the, perhaps erroneous, conclusion that we can use Mellin's Inverse Formula to calculate $$F_{H_x}(t)=\frac{1}{2\pi i}\lim_{r\to \infty}\int_{\gamma-ir}^{\gamma+ir}e^{\lambda t}\frac{e^{-x(-c+\sqrt{c^2+2\lambda})}}{\lambda} d\lambda,$$ where $\gamma$ can be any real number bigger than the real part of any poles of $\frac{e^{-x(-c+\sqrt{c^2+2\lambda})}}{\lambda}$ (so basically choose a favourable $\gamma>0$). Now my complex analysis is rather rusty, so I'm not entirely sure how to evaluate this integral. I think I might be able to use the Residue Theorem, but I'll need to refresh my memory. Before I do this though, I would like to verify whether this is the correct approach, or whether there is a quicker way to jump from $\mathbb E[e^{-\lambda H_x}]$ to $\mathbb P\{H_x<\infty\}$. Any comments on anyone of my statements or any part of my reasoning would be greatly appreciated.