Integrate $\int_{-\infty}^{+\infty}e^{-2x^2 + 7}dx$

Question

I need to evaluate the following integral

$$ \int_{-\infty}^\infty e^{-2x^2+7x} dx$$

I've tried u substitution, integration by parts, and splitting apart the exponent and don't seem to be making any progress.

Thanks for your help

Answer 1

Note that $$\int_{-\infty}^{\infty} e^{-2x^2 + 7x}dx=\int_{-\infty}^{\infty} e^{-2(x-7/4)^2}e^{49/8}dx$$ Then let $u=\sqrt{2}(x-\frac{7}{4})$ and recall $$ \int_{-\infty}^{\infty} e^{-u^{2}}du=\sqrt{\pi}.$$