Answer
$-\dfrac{1}{2}$
Work Step by Step
We are given that $\phi(x,y, z)=e^{-x} \cos y$
Next, the integral can be written for the given curve as:
$\int_C \nabla \phi \ dr=\phi(\ln 2 , 2\pi)- \phi (0, 0)\\=e^{-\ln 2} \cos (2 \pi)-e^{-0} \cos (0)\\=\dfrac{1}{2}-1\\=-\dfrac{1}{2}$
