Compute $\int \mathbf{F}\cdot d\mathbf{R}$ along the straight line $\mathbf{R}=t\mathbf{i}+t\mathbf{j}$ and the parabola $\mathbf{R}=t\mathbf{i}+t^2\mathbf{j}$, from (0,0) to (1,1). When $\mathbf{F}$ is a gradient field, use its potential $f(x,y)$.

$\mathbf{F}=(x\mathbf{i}+y\mathbf{j})/(x^2+y^2+1)$

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §15.2 Line Integrals, #34