Expectation of a function of Gamma random variable

Question

Consider a truncated exponential distribution $F(x\left| \lambda \right.) = \frac{{ - {e^{ - \lambda x}} + {e^{ - \lambda }}}}{{ - {e^{ - 2x}} + {e^{ - \lambda }}}}$ on the interval $[1,2]$. The parameter $\lambda (>0) $ is drawn from a Gamma distribution $\lambda \sim \Gamma (\alpha ,\beta )$ , with pdf $g(\lambda \left| {\alpha ,\beta } \right.) = \frac{{{\beta ^\alpha }}}{{\Gamma (\alpha )}}{\lambda ^{\alpha - 1}}{e^{ - \beta \lambda }}$, where $\Gamma (\alpha ) = \int_0^\infty {{z^{\alpha - 1}}{e^{ - z}}dz} $, $\alpha > 0$,$\beta > 0$. Try to obtain $\int_0^\infty {F(x\left| \lambda \right.)} g(\lambda \left| {\alpha ,\beta } \right.)d\lambda $.