So math stack exchange isn't really helping much with this. 
So initially, im proving the inverse laplace transform using contour integration.
I need to prove that:
$$L^{-1} \bigg[ \frac{1}{s}(\exp (- \sqrt{s/k}x)) \bigg] = \text{erf}(\frac{x}{2\sqrt{kt}})$$
This inverse laplace transform can be found using a table of Laplace Transform.
Using the following contour:

[enter image description here][1]

Then, after considering all contributions of this contour, I get:

$$L^{-1} \bigg[ \frac{1}{s}(\exp (- \sqrt{s/k}x)) \bigg] =
 1 - \frac{1}{\pi} \int_{0}^{\infty} \exp(-ut) \sin(\sqrt{u/k} x) \frac{du}{u}$$

Here, we can simplify the integral by letting: $v^{2} = ut$ and $y = x/\sqrt{kt}$ to get:
$$L^{-1} \bigg[ \frac{1}{s}(\exp (- \sqrt{s/k}x)) \bigg] = 1 - \frac{2}{\pi} \int_{0}^{\infty} \exp(-v) \sin(yv) \frac{dv}{v}$$.
How do I continue from here to eventually get to :
$$ 1 - \text{erf}(\frac{y}{2}) = 1 - \text{erf}(\frac{x}{2\sqrt{kt}}) $$
  [1]: https://i.sstatic.net/rnGle.png