I am not sure what you did, but you tought you can use the fundamental theorem of line integral you cannot.

Because the cross partials test fails.

The parametrization of the curve is,

x=t and y=2-t for 1<=t<=3.

That means,

INT(1,3)(x+2y)*(dx/dt)dt+INT(1,3)(x-2y)(dy/dt)dt

Insert the parametrization.

INT(1,3)(t+2(2-t))*(dt/dt)dt+INT(1,3)(t-2(2-t))*(dy/dt)dt

INT(1,3) 4-t dt + INT(1,3) 3t-4 dt

What ever that is should be the answer.