Final answer:
The line integral in question can be evaluated by applying Green's theorem and converting it into a double integral over the region enclosed by the curve. The key steps involve calculating the partial derivatives of the integrand's components and setting up the appropriate double integral.
Step-by-step explanation:
The line integral ∫C ex * cos(y) dx - ex * sin(y) dy can be evaluated using Green's theorem, which relates a line integral over a closed curve to a double integral over the region it encloses. Since the curve 'C' consists of two segments from A = (ln(2),0) to D = (0,1), and then D to B = (-ln(2),0), we should first understand that Green's theorem is applicable here because we can close the path by connecting B and A, thus enclosing a region.
To apply Green's theorem, we need to recognize that the line integral presented is equivalent to the circulation form where M(x,y) = ex * cos(y) and N(x,y) = -ex * sin(y). Therefore, we need to compute the partial derivatives ∂N/∂x and ∂M/∂y and then subtract them according to the theorem. As Green's theorem states, the line integral around a closed curve C is equal to the double integral over the region D enclosed by C of the expression (∂N/∂x - ∂M/∂y) dA.
For this specific problem, after finding the partial derivatives and setting up the double integral, we evaluate it over the region enclosed by the broken line and the straight line connecting B and A. If the region's geometry is simple enough, the integral can likely be done by hand, but for more complicated regions, a table of integrals or a computational tool might be necessary.