How to get rid of the integral in this equation $\int\limits_{x_0}^{x}{\sqrt{1+\left(\dfrac{d}{dx}f(x)\right)^2}dx}$? How to get rid of the integral $\int\limits_{x_0}^{x}{\sqrt{1+\left(\dfrac{d}{dx}f(x)\right)^2}dx}$ when $f(x)=x^2$?

Go back to the basics!
You need a derivative of the curve to begin with.
If the integrand has root of sum of squares in the denom
Immediately draw a right triangle with appropriate sides and the rest is an algebraic walk.
