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How would you integrate this?

$$ \int_{-\infty}^{\infty} \, e^{-ax^2} e^{-b(c-x)^2} dx$$

Where $a,b,c$ are constants.

I'm guessing you somehow get it to the form of $e^{-gx^2}$ , $g>0$

But not sure how to do this

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    $a,b,c$ are real constant and $a+b > 0$. Otherwise you'll need some complex analysis – 2017-02-25

1 Answers 1

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$$e^{-ax^2}e^{-b(x-c)^2} = e^{-ax^2 - b(x-c)^2)}$$

Now $-ax^2 - b(x-c)^2 = -((a+b)x^2 +2bcx + bc^2)$. Complete the square on this expression to convert it to the form $$-A(x+B)^2 + C$$for some $A, B, C$ which I will leave you to figure out. Your integral is then $$\int_{-\infty}^\infty e^{-A(x+B)^2 + C} \,dx \\=e^C\int_{-\infty}^\infty e^{-A(x+B)^2}\,dx\\=e^C\int_{-\infty}^\infty e^{-Au^2}\,du$$