Calculate the characteristic function $\varphi_W$ of W

Question

$p(x)=xe^{-x}$ for $x\geq 0$ or $0$ otherwise.

I tried to substitute $e^{-x}$ but then i found there is still a $x$ in front.

Answer 1

$$ \phi_W(t)=\langle e^{\mathrm{i}tx}\rangle=\int_0^\infty dx\ x e^{-x}e^{\mathrm{i}tx}=-\mathrm{i}\frac{d}{dt}\int_0^\infty dx\ e^{-(1-\mathrm{i}t)x}=-\mathrm{i}\frac{d}{dt}\frac{1}{1-\mathrm{i}t}=\frac{1}{(1-\mathrm{i}t)^2}\ . $$