Integrating equations using the integrating factor with higher powers of y

Question

I have often come across the following technique to integrate first order differential equations using the integrating factor: $$ \frac{dy}{dx}+P(x)y=Q(x) $$ $$ e^{{\int}P(x)dx}\frac{dy}{dx}+e^{{\int}P(x)dx}P(x)y=e^{{\int}P(x)dx}Q(x) $$ $$ \frac{d}{dx}\left(ye^{{\int}P(x)dx}\right)=e^{{\int}P(x)dx}Q(x) $$ And then simply integrate both sides with respect to $dx$. However I have never saw something like this for higher powers of y even if it should be possible under certain conditions. Like for example: $$ ny^{n-1}\frac{dy}{dx}+P(x)y^n=Q(x) $$ $$ e^{{\int}P(x)dx}ny^{n-1}\frac{dy}{dx}+e^{{\int}P(x)dx}P(x)y^n=e^{{\int}P(x)dx}Q(x) $$ $$ \frac{d}{dx}\left(y^ne^{{\int}P(x)dx}\right)=e^{{\int}P(x)dx}Q(x) $$ And then once again integrate both sides with respect to $dx$. Yet I have never seen a example of this in any textbook or formula booklet (unless I missed it). So I was wondering if this works or if I overlooked something or if there is a flaw in my reasoning? Is it simply that these conditions are rarely met and that's why I haven't come across a example?