The question

Consider the vector fieldF(x, y, z) = (yz + 1)i+ xzj+ (xy + $\displaystyle 2e^{2z}$)k.

Evaluate $\displaystyle \int_C{F.dr}$ along the path C from (0, 0, 0) to (1, 0, 0), following the helix (x, y, z) = (cos(t), sin(t), t) from (1, 0, 0) to (1, 0, 4$\displaystyle \pi$) and then the straight line from (1, 0, 4$\displaystyle \pi$) to (0, 0, 0).

My attempt

I first found a scalar potential $\displaystyle \phi$ with the property $\displaystyle \nabla{\phi}$ =F, which is $\displaystyle xyz + x + e^2z$

So the vector field is conservative. I noticed that the path is actually a closed loop, so if I'm not mistaken, I can apply the Fundamental Theorem of Line Integrals. Thus, the solution should be 0, right? However my book says -1. :/

Where have I gone wrong? Thanks.