Suppose ∇f(x,y)=6ysin(xy)i⃗+6xsin(xy)j⃗ , F =∇f(x,y) , and C is the segment of the parabola y=3x^2

Suppose ∇f(x,y)=6ysin(xy)i⃗+6xsin(xy)j⃗ , F =∇f(x,y) , and C is the segment of the parabola y=3x^2 from the point (3,27) to (5,75) . Then ∫C (F ⃗ ⋅dr) please help

First we find f(x, y).

∫ 6y sin(xy) dx = -6 cos(xy) + g(y)
∫ 6x sin(xy) dy = -6 cos(xy) + h(x).
==> f(x, y) = -6 cos(xy) + C.
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By the Fundamental Theorem of Calculus for Line Integrals, this equals
-6 cos(xy) {for (x, y) = (3, 27) to (5, 75)}
= -6 cos 375 + 6 cos 81.

I hope this helps!