Calculating endpoint for a function so arc length equals l. I'm trying to simulate a line hanging from a given point using a quadratic function. The line is located at point (x_0, y(x_0)) where y is my quadratic function.

Calculating endpoint for a function so arc length equals l.
I'm trying to simulate a line hanging from a given point using a quadratic function.
The line is located at point $(x_0,y(x_0))$ where y is my quadratic function.
Now, the line has length l, and I simulate it hanging and swinging from side to side by changing the quadratic and linear coefficient. This isn't really important in the problem, but I mention it just to give You a full picture.
Now, the problem is, that when I change the coefficients of the formula, the length of the line changes as well, and I have to pick a point $x_1$ in which the line will end so the length can stay the same.
Of course, the simplest way is just to pick a point $x_1$ such that distance from $(x_0,y(x_0))$ to $(x_1,y(x_1))$ is equal to l, but this actually calculates the straight-line distance, not the arc length, so for certain coefficeints this becomes really inacurate.
So, the proper way to find $x_2$ would be solving this equation:
$${\int <!--\; \int \; -->}_{x_0}^{x_1}\sqrt{1+f^{\prime }(t{)}^2}dt=l$$
For a proper $x_1$. However, after integrating it's really complicated to "extract" $x_1$ from the equation.
Thinking about this, I have found a stupid idea that doesn't work, but I dont know why, and this is the main point of my question.
So, for this integral exists a antiderivative F(x), so this whole equation can be represented as:
$$F(x_1)-<!--\; -\; -->F(x_0)=l$$
So:
$$F(x_1)=l+F(x_0)$$
Now, l is a constant, and $x_0$ is set (I'm just solving for $x_1$), so $F(x_0)$ is a constant as well. This means, that I just can differentiate the whole equation over $x_1$ and get:
$$F^{\prime }(x_1)=0=>\sqrt{1+f^{\prime }(x{)}^2}=0$$
Which is obviously wrong, because it doesn't depend at all on values of l and $x_1$.
So now, I have two questions: 1. Why is my differentiaton step wrong? 2. Is there any way to simplify solving for this integral?