Finding the inverse of the arc length equation of a parabola I need help in reversing the equation of the arc length of a parabola. 
$$y = \frac{1}{2} x \sqrt{1+4\cdot x^2} + \frac{1}{4} \cdot \ln(2\cdot x + \sqrt{1+4\cdot x^2})$$
Thank you.

This problem seem to somewhat misunderstood. Let me express it this way: Find the function $y(x)$ whose arc length is given by
$$s=\int_0^x\sqrt{1+\left(\frac{dy}{dx} \right)^2}~dx=\frac{1}{2} x \sqrt{1+4\cdot x^2} + \frac{1}{4} \cdot \ln(2\cdot x + \sqrt{1+4\cdot x^2})$$
Differentiating both sides we get
$$\sqrt{1+\left(\frac{dy}{dx} \right)^2}=\sqrt{4x^2+1}\\
1+\left(\frac{dy}{dx} \right)^2=4x^2+1\\
\frac{dy}{dx}=2x\\
\fbox{$y=\pm x^2+C$}
$$
I have verified this solution numerically.
