How to solve this equation with error functions? I'd like to solve the following equation for $S$, in terms of $p, x,$ and $\sigma$:

$$
p = \int_0^x\sqrt{\frac{2}{\pi}} \frac{e^{\frac{-(S-\mu)^2}{2\sigma^2}}}{\mathrm{erf}\left(\frac{S}{\sqrt{2}\sigma}\right)- \mathrm{erf}\left(\frac{S-1}{\sqrt{2}\sigma}\right)} d\mu
$$

Solving the integral as Mhenni suggested yields:
$$
p = \frac{\frac{\sqrt{2}}{2}\left(\mathrm{erf}\left(\frac{S-x}{\sqrt{2}\sigma}\right) - \mathrm{erf}\left(\frac{S}{\sqrt{2}\sigma}\right) \right)}{{\mathrm{erf}\left(\frac{S}{\sqrt{2}\sigma}\right)- \mathrm{erf}\left(\frac{S-1}{\sqrt{2}\sigma}\right)}}
$$

The problem now is dealing with the $S$'s inside the error functions. I know there's an inverse error function ($\mathrm{erf}^{-1}$), but I don't see how this is ultimately solvable for $S$.

Is this possible?

Thanks for any help!
I suppose that there are some typo's.

The result I obtained is $$p=\sigma\frac{  \text{erf}\left(\frac{S}{\sqrt{2} \sigma
   }\right)-\text{erf}\left(\frac{S-x}{\sqrt{2} \sigma
   }\right)}{\text{erf}\left(\frac{S}{\sqrt{2} \sigma
   }\right)-\text{erf}\left(\frac{S-1}{\sqrt{2} \sigma }\right)}$$ I do not think that you could be able to solve analytically the equation for $S$. More than likely, only numerical methods could do it.

If you provide some typical values for $p,\sigma,x$, I could look at some numerical scheme.
You have not changed to limits of integration 


  $$
p = \frac{ \sqrt{\frac{2}{\pi}} }{   {\mathrm{erf}\left(\frac{S}{\sqrt{2}\sigma}\right)- \mathrm{erf}\left(\frac{S-1}{\sqrt{2}\sigma}\right)}} 
   \int_{S/(\sqrt{2}{\sigma}) }^{  (S-x) /(\sqrt{2}{\sigma}) } e^{-t^2} dt. 
$$


Added: You need my answer to finish evaluating the integral.