Some expectation values for a Gamma distribution Assuming I have a Gamma distributed random Variable $x \sim Gamma( \alpha, \beta )$. Now I like to have the following two expectation values (integrals):


*

*$E \left[ x \ln x \right]$

*$E \left[ \ln \Gamma \left( x \right) \right]$
with $\Gamma \left( x \right)$ being the Gamma function
Many thanks in advance
 A: First integral. $X$ has density function 
$$
f(x;\alpha ,\beta ) = \frac{{\beta ^\alpha  }}{{\Gamma (\alpha )}}x^{\alpha  - 1} e^{ - \beta x},\;\; x > 0, 
$$
with $\alpha,\beta > 0$ fixed.
It follows readily from
$$
\Gamma '(\alpha ) = \int_0^\infty  {x^{\alpha  - 1} e^{ - x} \ln x\,{\rm d}x} 
$$
and
$$
\Gamma '(\alpha ) = \Gamma (\alpha )\psi _0 (\alpha ),
$$
where $\psi_0$ is the  digamma function,
that 
$$
{\rm E}[X\ln X] = \frac{{\beta ^\alpha  }}{{\Gamma (\alpha )}}\int_0^\infty  {(x\ln x)x^{\alpha  - 1} e^{ - \beta x} dx} = \frac{\alpha }{\beta }[\psi_0 (\alpha  + 1) - \ln \beta ].
$$
EDIT: Elaborating (in response to the OP's request). 
First, you can find here the above formulas for $\Gamma'(\alpha)$. Now, using a change of variable, we have
$$
\Gamma'(\alpha+1) = \int_0^\infty  {\beta ^{\alpha+1}  x^{\alpha} e^{ - \beta x} \ln (\beta x)\,{\rm d}x} =  \ln \beta \int_0^\infty  {\beta ^{\alpha+1} x^{\alpha} e^{ - \beta x} \,{\rm d}x}  +  \int_0^\infty  { \beta ^{\alpha+1} x^{\alpha} e^{ - \beta x} \ln x\,{\rm d}x}. 
$$
The first integral on the right-hand side is equal to $\Gamma(\alpha+1)\ln \beta$, and the second integral to $\Gamma(\alpha)\beta {\rm E}[X\ln X]$. It thus follows that
$$
\Gamma (\alpha  + 1)\psi _0 (\alpha  + 1) = \Gamma (\alpha  + 1)\ln \beta  + \Gamma (\alpha )\beta {\rm E}[X\ln X].
$$
Finally, by $\Gamma(\alpha+1)=\alpha \Gamma(\alpha)$, we get
$$
{\rm E}[X\ln X] = \frac{\alpha }{\beta }[\psi_0 (\alpha  + 1) - \ln \beta ].
$$
A: 1.
E(XlnX)=(formula of expectation, group x) =E(lnY) with Y~gamma(alpha+1,beta)
E(lnY)=ψ(alpha+1)-ln(beta) where ψ(k) is the digamma function.
