integrating this infinite gaussian integral How does one integrate 
$\int_{-\infty}^{+\infty}x e^{-\lambda ( x-a )^2 }dx $
where $\lambda$ is a positive constant.
My integral tables are not returning anything useable. The best it return is non-definite gaussian integrals. Useless!
Help please

$$
\begin{align}
\int_{-\infty}^{+\infty}x e^{-\lambda(x-a)^2}\,\mathrm{d}x
&=\int_{-\infty}^{+\infty}(x+a) e^{-\lambda x^2}\,\mathrm{d}x\tag{1}\\
&=\frac{a}{\sqrt{\lambda}}\int_{-\infty}^{+\infty}e^{-x^2}\,\mathrm{d}x\tag{2}\\
&=a\sqrt{\frac\pi\lambda}\tag{3}
\end{align}
$$
Explanation:
$(1)$: substitute $x\mapsto x+a$
$(2)$: $\int_{-\infty}^{+\infty}xe^{-\lambda x^2}\,\mathrm{d}x=0\because$ odd integrand, then substitute $x\mapsto\frac x{\sqrt\lambda}$
$(3)$: $\int_{-\infty}^{+\infty}e^{-x^2}\,\mathrm{d}x=\sqrt\pi$
