I have a problem of estimating moments of an exponential family by integration by parts. Lets consider the exponential family in its canonical form. $f(x)=e^{\theta x-\psi(\theta)}h(x)$.
The parameter is $\theta$. $x$ is univariate.
$\psi(\theta)$ is the normalizing constant.
I want to compute $E(x)$ by definition (the expected value of $x$).
What I know
I know that for exponential families $E(x^k)=\psi^k (\theta)$. Where $\psi^k $ represents the $k$th derivative of $\psi$.
Hence $E(x)=\psi^\prime(x)$.
To get $E(x)$ the proof starts by:
$\int_{-\infty}^{\infty}f(x)dx=1$ (because $f(x)$ is a probability distribution) then differentiate both sides with respect to $\theta$ and more manipulations are done to get $E(x)=\psi^\prime(x)$
What I will like to be helped on.
Use the definition of the expected value to arrive at the same solution $E(x)=\psi^\prime(x)$.
That is show that $E(x)=\int_{-\infty}^{\infty}xf(x)dx=\psi^\prime(x)$.
I have tried to use integration by parts. But I get a $0$.
Can someone nicely use integration by parts to solve this.