Let and let C be the path joining (0,0) to (1,2) in the plane. Evaluate
I know that the it isn't conservative but other than that, I don't know what to do after that.
Remember that
Where is the unit tangent vector. Since you given a function for your path we can parametrize it trivially
So in your case we get
Now recall that tangent vector is given by
Now to normalize it we get
and the arc length
Combining all of this gives