# Expectation of $h \circ X$

I'm only starting to learn statistics.

The definition I've been given for the expected value (expectation) of a continuous random variable X with probability density function (PDF) $$f_X$$ is the following

$$E[X] = \int_{-\infty}^{+\infty}f_X(x)xdx$$

Despite only beginning stats, I've already stumbled many times on the use of a property that was not explicited and that I don't find any explanation for. I don't know the exact conditions of application of this property, but it seems that $$E[h(X)]$$, where h is a function from $$\mathbb{R}$$ to $$\mathbb{R}$$, that, I guess, has to verify some conditions, is often expressed, by some people as:

$$\int_{-\infty}^{+\infty}h(x)f_X(x)dx$$

For example, in the course I'm following, they define the variance $$Var(X)$$ as the expectation of $$(X - \mu)^2$$. All I get from this definition is that if $$f_{(X-\mu)^2}$$ is the PDF of $$(X - \mu)^2$$, then

$$Var(X) = \int_{-\infty}^{+\infty}f_{(X-\mu)^2}(x)xdx$$

Yet, without other explanation, the variance is replaced with the following expression:

$$Var(X) = \int_{-\infty}^{+\infty}(x-\mu)^2f_X(x)dx$$

during the course. Other example, in this post:

https://stats.stackexchange.com/a/133474/67507

it is written that ($$f$$ being the PDF of $$X$$): $$E[X^2] = \int_{-\infty}^{\infty}{x^2 f(x) dx}$$

Integration is also a work in progress for me at the moment, so probably I'm just missing some basic property of integrals to make this step from $$\int_{-\infty}^{+\infty}f_{h \circ X}(x)xdx$$ to $$\int_{-\infty}^{+\infty}h(x)f_X(x)dx$$ But what is the theorem/property that allows to do that and where can I find a proof of it?