This writeup states the accepted and assumed definition of the integral of a complex function. I feel it is worth noding since, unlike the calculus of real variables, the calculus of complex variables was developed merely for the entertainment of mathematicians, so the definition was a bit arbitrary. However, the development of complex integration led to beautiful mathematics, and (surprisingly to me) very useful techniques for physicists and engineers.
Definition: The integral of a complex function, f(z) = u(z) + iv(z), where z = x + iy, is given by lim (as Δz goes to 0) Σf(z)Δz. Of course the Δz's comprise the path between the endpoints of some curve in complex space and f(z) is evaluated at any point inside each Δz.
The definition can be expanded into integral (udx - vdy) + i*integral (udy + vdx), and it is readily apparent that the definition is different from that of 2-d line integrals such as the integral of a force between two points, which corresponds to energy. One thing to note is that, in general, the integral is not purely a function of the endpoints, but rather a function of the path between the endpoints. The condition for independence of path (a little bit analogous to the definition of a conservative force in physics) is that that the function f(z) is analytic in the region on and inside the region enclosed by the two paths connecting the endpoints (see Cauchy-Goursat Theorem). However, if two paths between the endpoint enclose a singularity, the integrals need not be equal.