Solution
= Solution
The <fundamental theorem for line integrals> evaluates this <conservative vector field> integral by the change in potential. At the initial point $\phi=0$. At the final point $z=0$, so the exponential factor is one, and
$$
\boxed{\int_C\mathbf F\cdot d\mathbf r
=\phi(C_{\rm end})-\phi(C_{\rm start})
=\sin\frac\pi8\cos\frac\pi4
=\frac{\sqrt2}{2}\sin\frac\pi8}.
$$
The detailed shape of the path does not affect this <line integral>.