Let $w(x) = x \log{x}$
$x \sim Gamma(\alpha = 3.7, \lambda = 1)$
Find $E[w(x)]$
I have set up the following integral:
$\int_0^{\infty} x\log{x} \frac{\lambda^{\alpha}}{\Gamma(\alpha)} x^{\alpha -1} e^{-(\lambda)x}dx$
Brute-forcing it doesn't seem to be working, but I can't find the "trick".
I know from simulation that it should be approximately 5.32, I just need to compare it to the analytical solution.