Evaluate the line integral
$$ \int_C (\ln y) e^{-x} \,dx - \dfrac{e^{-x}}{y}\,dy + z\,dz $$
where C is the curve parametrized by $r(t)=(t-1)i+e^{t^4}j+(t^2+1)k$ for $0\leq t\leq 1$
I know that the potential function is $$f(x, y, z) = -e^{-x}\ln y + \frac{z^2}{2} + C$$ but exactly how would I use that to evaluate the line integral?
Pretty much like an ordinary real integral. The potential function takes the place of the antiderivative; if the the path goes from $A$ to $B$ then the integral is $$f(B)-f(A)\ .$$ In this case $A$ is $r(0)$ and $B$ is $r(1)$. Can you do the calculations?