Calculate the line integral of F along the curve C.?

Close the contour by connecting Q to P. Call the new contour C', then use Green's theorem to conclude that

∫ F•dr = ∫ ∫ (∇xF)•k dA
C' . . . . .R

where R is the region enclosed by the new contour.

Fortunately, (∇xF)•k = 0. So your integral reduces to

P
-∫ F•dr
Q

Parametrize the line from P to Q (flip limits to absorb the minus sign) as

x = 3π/2 and y = t - 3π/2 for 0 ≤ t ≤ 3π.

The integral becomes

3π
∫ 5/√(2) cos(t/2 - 3π/4) dt = 10.
0