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 $0\leq t\leq 1$
2026-03-26 07:55:21.1774511721
line integral (multivariable calculus)
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By my previous answer, the vector field: $$ \vec F = \left\langle e^{-x} \ln y, \frac{-e^{-x}}{y}, z \right\rangle $$ is conservative. Thus, the exact path of integration is irrelevant and we only care about the endpoints. Simply use the potential function from my previous answer and apply the Fundamental Theorem for Line Integrals.