# Integrating a random variable with known density

If an application $f$ has a known density (for example, $f\sim\mathcal{N}(0,1)$), does the following equality stand : $\int_{-\infty}^{+\infty}f(x) \; \textrm{d}x = \int_{-\infty}^{+\infty}f(x) \; \textrm{pdf}(f(x)) \; \textrm{d}x$ ?

If yes: how to prove this ? If not: is it possible to quantify the error which is introduced ?

-
Can you express this in more standard probability language? – André Nicolas Nov 12 '12 at 20:26