I am new to calculus. Today when I read the exponential family, The exponential family are defined as below:
$$ p(x|\alpha) = h(x)exp\{ \alpha T(x) - A(\alpha)\}$$
$ T(x) $ is referred to as sufficient statistics. and $A(\alpha)$ satisfied the formular in the bottom.
there is one formular which I did not understand which is:
$$ \int T(x)exp\{ \alpha T(x) \}h(x)dx \over \int exp\{ \alpha T(x) \}h(x)dx $$ $$ = \int T(x)exp\{ \alpha T(x) - A(\alpha) \}h(x)dx $$
where $ A(\alpha) = log \int h(x)exp\{ \alpha T(x) \}dx $
I am quite confused why how does this come out? Is there any theorem for the division of two integrations?