Length of an arc

I've been given the question: taking the line

$$Y=x^2$$
Find the length of the line between

$$X=1,x=2$$

I've been given the working but I don't understand it, I know that once you find $Dx/dy^2$ that you have to use a hyperbolic function but I don't know why. Then following this all the way down I don't understand the different stages. For example there's one point where there's a

$$ 1+\sin^2(h)^{1/2} $$
or something like that but I've no idea where the $1/2$ came from. I know its a bit of a pain in the butt, but would someone be able to explain the full process for me please?

P.s.s sorry for the terrible editing of this post I'm doing it on mobile app for the first time.

Answer

In cartesian coordinates, the arc length is given by $$L=\int_a^b\sqrt{1+\left(\frac{dy}{dx}\right)^2}\;dx$$ If $y=x^2$, the problem is then to compute first $$I=\int\sqrt{1+4x^2}\;dx$$ Changing variable $x=\frac{\sinh (t)}{2}$, $dx= \frac{\cosh (t)}{2}\,dt$ $$I=\frac 12\int\cosh ^2(t)\;dt=\frac 14\int (1+\cosh(2t))\;dt=\frac 14(t+\frac 12 \sinh(2t))$$ and back to $x$ $$I=\frac{1}{2} x\sqrt{1+4 x^2} +\frac{1}{4} \sinh ^{-1}(2 x)$$