Exact Equations & Substitutions · Superchallenge problem

The solution of the Bernoulli equation $y' = y - y^3$ with $y(0) = \dfrac12$ increases toward $1$ as $x \to \infty$, so $\displaystyle\int_0^x y(t)\,dt$ grows like $x$ minus a lag. Find $\displaystyle\lim_{x\to\infty}\left(x - \int_0^x y(t)\,dt\right)$.