Expected value (exp(x)-1)^a

Hello,

I love your channel and your community. Thanks a lot for your work!

I'm stuck with an integral since a while and I'm getting obsessed :)

integral_0^infinity (exp(x)-1)^a exp (-1/2 * ((x-mu)/sigma)^2) dx

0<alpha<1

(expected value of (exp(x)-1)^alpha where x follows on normal distribution mu sigma)

I've managed to find an approximation (see attachment) and it's sufficient for my need but I feel like the exact solution exists. I've learned a lot of new techniques when trying to solve this integral :)

Do you have some tips?

Thanks,

Regards,

Mathieu