Answer
$-\sqrt 2$
Work Step by Step
Since, we have $ds=\sqrt{(\dfrac{dx}{dt})^2+(\dfrac{dy}{dt})^2+(\dfrac{dz}{dt})^2} dt$
or, $ds=\sqrt{1^2+(-1)^2+0^2} dt \implies ds= \sqrt 2 dt$
Now, the line integral is: $\int_C (x-y+z-2) ds=\int_0^1 (t-(1-t)+1-2) \sqrt 2 dt$
or, $=\sqrt 2\int_0^1 (2t-2) dt$
or, $\sqrt 2(1-2)=-\sqrt 2$
