Integrating Gaussian type integrand

Question

Any idea how to perform these two integrations?

1. $$ \int_{0}^{\infty} \frac{exp(-a x^2)}{x^2(x^2+\kappa^2)} dx $$ 2. $$ \int_{0}^{\infty} \frac{exp(-a x^2)}{(x^2+\kappa^2)} dx $$ second integration is same as equation 1.42 of this link

Answer 1

I am afraid that these integrals do not show explicit expression.

Considering the second one $$I=\int \frac{e^{-a x^2}}{x^2+\kappa ^2}\,dx$$ let $x=\kappa y$ and $b=a \kappa ^2$ to make $$I=\frac 1 \kappa \int \frac{e^{-b y^2}}{y^2+1}\,dy=\frac1 \kappa\int e^{-by^2}\Big[\sum_{n=0}^\infty (-1)^n y^{2n}\Big]\,dy$$ $$I=\frac1 \kappa\sum_{n=0}^\infty (-1)^n\int y^{2n}\, e^{-b y^2}\,dy=\frac1 \kappa\int e^{-b y^2}\,dy+\frac1 \kappa\sum_{n=1}^\infty (-1)^n\int y^{2n}\, e^{-b y^2}\,dy$$ with $$\int e^{-b y^2}\,dy=\frac{\sqrt{\pi } }{2 \sqrt{b}}\,\text{erf}\left(\sqrt{b} y\right)$$ $$\int y^{2n}\, e^{-b y^2}\,dy=b^{-(n+\frac{1}{2})} \Gamma \left(n+\frac{1}{2},b y^2\right)$$