Integrals of form $\int^{∞}_{-∞} x^{2k} \exp(-b^2 (x+x_0)^2)dx$, $k=1,2$. Please can anybody help me to solve these integrals:

$$\int^{∞}_{-∞} x^2 \exp(-b^2 (x+x_0)^2)dx\,,\,\,\int^{+∞}_{-∞} x^4 \exp(-b^2 (x+x_0)^2)dx\;\;?$$


One may start with the gaussian integral
$$
\int^{∞}_{-∞} \exp(-b^2 u^2)du=\frac{\sqrt{\pi}}{b},\qquad b>0,
$$ getting, by differentiation with respect to the parameter $b$,
$$
\int^{∞}_{-∞} u^2\exp(-b^2 u^2)du=\frac{\sqrt{\pi}}{2b^3},
$$$$
\int^{∞}_{-∞} u^4\exp(-b^2 u^2)dx=\frac{\sqrt{\pi}}{4b^4}.
$$ Then, by the change of variable
$$
u=x+x_0,\qquad du=dx,
$$ one obtains
$$\int^{∞}_{-∞} x^2 \exp(-b^2 (x+x_0)^2)dx=\int^{∞}_{-∞} (u-x_0)^2\exp(-b^2 u^2)du
$$$$\int^{∞}_{-∞} x^4 \exp(-b^2 (x+x_0)^2)dx=\int^{∞}_{-∞} (u-x_0)^4\exp(-b^2 u^2)du
$$ and one may conclude by expanding the integrand, using the parity and the above results.
