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Translating for continuous random variables

I'm also aware of the result ∫ P(x).x dx

In my lecture notes ( I more or less transcribed from what the lecturer said ):

∫ P(x).x^2 dx = <x^2> , should it not be ∫ P(x^2).x^2 dx = <x^2>?

Does P(x^2) even mean anything in relation to P(x) ? I find it difficult to link the two.

EDIT: Touching on the ∫ P(x^2).x^2 dx = <x^2> confusion again, for ∫ P(x^2).x^2 dx = <x^2> would you have to be integrating wrt (x^2) too? Ahh, much confusion.

Could somebody please clear this up for me? Any examples would be much appreciated