On Wikipedia, there is an alternative form of the integral of the Gaussian function, which is $$\begin{align} \int_{-\infty}^{\infty} k\exp\left(-fx^2 + gx + h \right)dx &=\int k\exp\left( (-f(x-g/(2f))^2 + g^2/(4f) + h\right)dx \\ &= k\sqrt{\frac{\pi}{f}}\exp\left(\frac{g^2}{4f}+h\right) \end{align}$$ For the case where $gx+h=0$, this can be solved using polar coordinates. For $h=0$, I can complete the square. For the case where neither are zero, I'm not sure what to do.

  • $\begingroup$ Can you not also complete the square if $h \neq 0$? $\endgroup$ Commented Jan 6, 2021 at 9:45
  • 2
    $\begingroup$ $e^{\frac {g^{2}} {4f}+h}$ is a constant. Pull it out and make the substitution $y=x-\frac g {2f}$. $\endgroup$ Commented Jan 6, 2021 at 9:49
  • $\begingroup$ @KaviRamaMurthy thanks, i did not notice that right away. $\endgroup$ Commented Jan 6, 2021 at 9:53


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