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:

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


An idea that you shall complete: substituting in first integral
$$u:=x+x_0\implies du=dx\implies \int_{\Bbb R}x^2e^{-b^2(x+x_0^2)}dx=\int_{\Bbb R}(u-x_0)^2e^{-(bu)^2}du=$$
$$=\int_{\Bbb R}u^2e^{-(bu)^2}du-2x_0\int_{\Bbb R}ue^{-(bu)^2}du+x_0^2\int_{\Bbb R}e^{-(bu)^2}du$$
The first integral above is doable by parts or differentiating under the sign integral, the second one is immediate as $\;u=(-2bu)\cdot\left(-\frac1{2b}\right)\;$, and the last one is a usual Gaussian integral
