When I think about my school years, I don't think I've ever encountered a problem where I had to isolate integrated variables. Could anyone guide me through how I should solve for $x$ in the following integral? This is the integral describing the Gaussian error function $\text{erf}(x)$.
$\frac{2}{\sqrt{\pi}}\int\limits_{0}^{x}e^{-t^2}dt = 0.7$
So, you want to solve $$\text{erf}(x)=k$$
There is no explict solution and numerical methods should be used.
However, you can get a quite good approximation using $$\text{erf}(x)\approx \sqrt{1-\exp\Big(-\frac 4 {\pi}\,\frac{1+\alpha\, x^2}{1+\beta \,x^2}\,x^2 \Big)}$$ where $$\alpha=\frac{10-\pi ^2}{5 (\pi -3) \pi } \qquad \text{and} \qquad \beta=\frac{120-60 \pi +7 \pi ^2}{15 (\pi -3) \pi }$$ which can easily be inversed (at the price of a quadratic in $x^2$) (have a look here).
Applied to your case, this would give $x=0.732883$ while the exact solution would be $x=0.732860$.
$x$ is equivalent to $\operatorname{erf}^{-1}(0.7)$. The Maclaurin series for $\operatorname{erf}^{-1}x$ shows us analytically that $x=u+\frac{1}{3}u^3+\frac{7}{30}u^5+\ldots$, where $u=0.7\frac{\sqrt{\pi}}{2}$ and the coefficients of the series are given by a recurrence relation at Mathworld: Inverse Erf. However, it seems unlikely that a closed form for $x$ would be known as it is not even yet known whether $\operatorname{erf}^{-1}(0.5)$ has a closed form.
