Suppose $F = (P,Q)$ is a vector field that is irrotational so that $\text{rot} (F) = Q_x - P_y = 0$.
Then, show that $F$ is conservative with a potential function of $$f(x,y) = x \int_0^1{P(tx, ty) dt} + y\int_0^1{Q(tx,ty)dt}.$$
We can suppose that $F$ is defined on a convex domain including the origin.
What I've tried:
I want to differentiate $f$ with respect to $x$ and hopefully recover $P$. So, product rule yields $$\frac{\partial f}{\partial x} = \int_0^1P(tx, ty)dt + x\int_0^1{tP_x(tx,ty)dt}+y\int_0^1{tQ_x(tx,ty)dt}.$$
If $F$ has rot $0$, then $Q_x = P_y \implies$
$$\frac{\partial f}{\partial x} =\int_0^1P(tx, ty)dt + x\int_0^1{tP_x(tx,ty)dt}+y\int_0^1{tP_y(tx,ty)dt}.$$
I'm not sure where to proceed from here.
Good. Now group the last two terms as $$\int_0^1 t\big(xP_x(tx,ty)+yP_y(tx,ty)\big)dt = \int_0^1 t\frac d{dt}P(tx,ty)\,dt.$$ Can you finish now?