Using the normalization integration for a Gaussian random variables,find an analytic expression(closed-form solution) for the following integral
$I=\int^{\infty}_{-\infty}e^{(-(ax^2+bx+c))}dx$,where $a \gt 0,b $ and $c$ are constants
There is a Hint below the question,but i don't know how to use this hint to calculate this question
Hint: Use the Gaussian integration $\frac{1}{\sigma \sqrt{2 \pi}}\int^{\infty}_{-\infty}e^{- \frac{1}{2} \frac{(x-m)^2}{\sigma ^2}}dx=1$
You can write: $$ a x^2 +b x +c= a \left(x+\frac{b}{2a} \right)^2+c-\frac{b^2}{4a}$$ So by linearity of the integral: $$I=e^{-c+\frac{b^2}{4a}} \int_{-\infty}^{+\infty}e^{-\frac{1}{2}\frac{(x+\frac{b}{2a})^2}{1/(2a)}} dx $$ which is exactly of the form needed.