integration - distribution of constants over integrals

I'm seeing this as part of a proof for the reduction formula and I see this:

So am I correct for saying that if you multiply the $sin^{n-2}{x}$ by $(1 - sin^2{x})$, you get $sin^{n-2}{x} - sin^n{x}$ and so the $(n-1) \int sin^{n-2}{x}\cdot cos^2{x}$ becomes what is shown below? is that right? Generally, $\int a - b = \int a - \int b$ right? That integral rule is visually intuitive right?