How can we simplify this integral? $\int{x\frac{f(x)}{\int f(x) dx} dx}$

It should read$$\int x\frac{f(x)}{\int_a^bf(y)dy}dx=\frac{\int xf(x)dx}{\int_a^bf(y)dy},$$where the $x$-integral may or may not be definite (though in this context it would be).

I think maybe it is meant to read something like: $$\int_a^bx\frac{f(x)}{\int_a^bf(x)dx}dx$$ as then the bottom integral is a constant and you can bring it outside: $$\frac{1}{\int_a^bf(x)dx}\int_a^bxf(x)dx=\frac{\int_a^bxf(x)dx}{\int_a^bf(x)dx}$$

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Or maybe if it is: $$\int x\frac{f(x)}{\int f(x)dx}dx$$ then you can define: $$F(x)=\int f(x)dx$$ so you get: $$\int x\frac{F'(x)}{F(x)}dx=\int\frac{F^{-1}(u)}{u}du$$ I don't really understand the notation but its a suggestion