First Order Differential Integration (Integrating Factor?)

Hey, I was wondering whether anyone could help me with the first part of a question I am stuck on, integrating:
$xy (dy/dx) = 1$
I would assume that I have to find an integrating factor, however to do this we have been taught to get the equation into the form
$(dy/dx) + P(x)y = Q(x)$
However the closest I can get is
$(dy/dx) - (1/xy) = 0$
This gives me an integrating factor of -ln x if I just assume that y^-1 works the same as y however I assume that it doesn't as this doesn't seem to help. Any advice?

Aha I did think that, however we've never really been taught WHEN an equation is separable, I don't really like the idea of just 'multiplying' across dx and my physics teacher says this doesn't always work. But thanks

I don't know why it works either, but at A-level, you can safely assume that it does