Mean Residual Life

#1
Hi,

I've been tasked with solving the mean residual life given the density function f(x) and the survival function S(x)

I think mean residual life can be calculated by:
mrl(x)=[∫(t-x)f(x)dt]/S(x)]

or

mrl(x)=∫S(t)dt/S(x)

the problem is, f(x) contains the gamma function and I don't know how to integrate these.

f(x)= [x^(αt) * exp(-x/λ)] / [(λ^α)(Γ(α)]

S(x)=exp[(λ/α) (1-e^(αx))]

any help would be most appreciated.