I am unsure how to calculate the following definite integral: $$ \int_0^\infty e^{-x^2}(x + k)^{\alpha}dx,$$ where $k > 0$ and $\alpha$ is a real number. I tried integrating by parts and also a few things with the Gamma function, but I am not getting anywhere. Maybe some slick trick with contour integration? Any help will be highly appreciated, thanks!
2026-04-02 19:46:40.1775159200
Improper definite integral $\int_0^\infty e^{-x^2}(x + k)^\alpha dx$
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