Why doesn’t this integration method work?

Question

Taking dy/dt = xy

Of course you can divide y from both sides getting a logarithmic function, but why can’t you integrate both sides with dt and get the right answer? Or what is wrong with this example?

Ex: ∫(dy/dt)dt=∫(xy)dt

The dt cancel out so ∫dy=xy∫dt

y=xyt divide both sides by y

1=xt

Answer 1

You know that $y$ (and probably $x$) is a function of $t$ so the step $$ \int xy\,dt = y\int x\,dt $$ is not valid.