How can I solve this definite Integral? In Bayesian Analysis

Question

This integral is part of an exercise of Bayesian analysis, but i don't know how to integrate I think completing a normal distribution can be helpful

Thanks in advance

Answer 1

Hints:

After expansion of the exponent and completing the square, you reduce to

$$\int_{-\infty}^\infty e^{-p(a-q)^2+r}da=e^r\int_{-\infty}^\infty e^{-pa^2}da$$

which is a familiar Gaussian integral.

Answer 2

Note that if you expand the exponent it is in the form $$x_1a^2 + x_2a + x_3$$

Also note that $$\int_{-\infty}^{\infty} e^{-ax^2\,+\,bx\,+\,c}\,dx = \sqrt{\frac{\pi}{a}}\,e^{\frac{b^2}{4a}\,+\,c}$$