= Solution
The incompressibility equation turns the advective term into a divergence:
$$
u u_x+v u_y=\partial_x(u^2)+\partial_y(uv),
$$
because $v_y=-u_x$. For a plume in an otherwise stationary environment, the boundary-layer pressure equals the ambient pressure and has no imposed $x$-gradient. Integrating the vertical momentum equation across $y$ therefore gives
$$
\frac{d}{dx}\int_{-\infty}^{\infty}\rho u^2\,dy
+[\rho uv]_{-\infty}^{\infty}
=\mu[u_y]_{-\infty}^{\infty}+b(x).
$$
Both boundary terms vanish as the velocity and shear decay away from the plume. Hence the <integral momentum-flux balance for a two-dimensional plume> is
$$
\boxed{\frac{d}{dx}\int_{-\infty}^{\infty}\rho u^2\,dy=b(x)}.
$$
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