Integrating exponential pdf for mgf

Question

I would like to integrate the following function but unfortunately I am not getting to the right outcome

$$\int_0^{\infty} e^{tx}\frac 16\left(e ^{-\frac x2}+xe^{-\frac x2}\right)$$

Answer 1

Consider $h(t)=\int_{0}^{\infty}e^{tx}e^{-\frac{x}{2}}\mathrm{d}x$, then $$ h(t) =\int_{0}^{\infty}e^{-x\left(\frac{1}{2}-t\right)}\mathrm{d}x =\frac{2}{1-2t} $$ Now observe that $\int_{0}^{\infty}xe^{tx}e^{-\frac{x}{2}}\mathrm{d}x=h'(t)=\frac{4}{(1-2t)^{2}}$

and so the integral you seek equals $\frac{1}{3}\left(\frac{1}{1-2t}+\frac{2}{(1-2t)^{2}}\right)$