There are three approaches to the evaluation of line integrals.

1--Parametrize a specific path and substitute into the integrand.

2--Determine that the "integrand" is curl-free on a simply-connected domain, seek a potential whose gradient is the integrand, and compute the difference of the potential at the end-points of the path.  (If you can find a potential for the integrand, it is not necessary to show the integrand is curl-free.)

3--Determine that the path C is closed and that C is the boundary of a region D.  Then compute the curl of the integrand and use Green's Theorem (in 2-space) or Stokes' Theorem (in 3-space) to convert the line integral to an area integral (or surface integral).