electrostatics - How to find the lines of force from electric field?

I know that the line of force at any point in electric field is tangent to the field there however how can I quantitatively find it? As an example if the electric field in a region is given by $\vec E=x\hat i+y\hat j$ how can I find the equation of line of force?

Answer

Suppose that at position $A$ $(x,y)$ the electric field is $\vec E = E_{\rm x} \hat i + E_{\rm y} \hat j$.

You want to move to a new position $B$ $(x+dx,y+dy)$ such that line $AB$ is along the same direction as the electric field vector at position $A$ which means that $\dfrac{dy}{dx} = \dfrac {E_{\rm y}}{E_{\rm x}}$.

In your example you do not have to do an integration.