Marginal distribution from joint distribution

Question

I have the joint distribution of two random variables $x,\theta$ $$h(x,\theta)=\frac{1}{2\pi\sigma}\exp\left\{-\frac{1}{2}\left[(x-\theta)^2+\frac{\theta^2}{\sigma^2}\right]\right\}$$ To find the marginal distribution of $x$, I need to integrate this over all possible values of $\theta$, so that $$m(x)=\frac{1}{2\pi\sigma}\int_{-\infty}^{\infty}\exp\left\{-\frac{1}{2}\left[(x-\theta)^2+\frac{\theta^2}{\sigma^2}\right]\right\}d\theta$$ I am kinda confused how to proceed from here, I'm sure there is some minor transformation that needs to be done but it is not clicking. Hints will be appreciated.

Answer 1

Hint

Start writing $$A=\frac{1}{2}\left[(x-\theta)^2+\frac{\theta^2}{\sigma^2}\right]=\frac{1}{2} \left(1+\frac{1}{\sigma ^2}\right)\theta ^2- x\,\theta+\frac{x^2}2 $$ Complete the square and use a clear substitution.

Yous should face a very simple integral and result