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I was trying to compete the following integral $$ \int_{0}^{\infty}x\exp\left(-\,{x^{2} \over 2}\right)\sin\left(sx\right)\,{\rm d}x $$

Felix Marin
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The game
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    Hi! Please provide context for your question: What have you tried? Where does the problem come from? Otherwise, your question will be likely to attract downvotes. – Robert Lee Oct 03 '22 at 21:43
  • This question came from attempting fourier inverse to PDE equation, I have tried to write things in term of complex variable but it is not going anywhere – The game Oct 03 '22 at 21:52
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    @rasheda564 please put the context in your post. Thank you. – Aaron Hendrickson Oct 03 '22 at 22:11

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We know that $\int_{0}^{\infty} e^{-x^2} \cos(ax)\mathrm{d}x = \frac{\sqrt{\pi}}{2}e^{-\frac{a^2}{4}}$. Recalling Feynman's trick we get \begin{align} \int_0^{\infty}xe^{\frac{-x^2}{2}}\sin(sx)\,\mathrm{d}x &=-\frac{\mathrm{d}}{\mathrm{d}s}\int_0^{\infty}e^{\frac{-x^2}{2}}\cos(sx)\,\mathrm{d}x \\ & \overset{x \to \sqrt{2}x}{=}-\sqrt{2}\frac{\mathrm{d}}{\mathrm{d}s}\int_0^{\infty}e^{-x^2}\cos(\sqrt{2}sx)\,\mathrm{d}x \\ & = -\sqrt{2}\frac{\mathrm{d}}{\mathrm{d}s}\left(\frac{\sqrt{\pi}}{2}e^{-\frac{s^2}{2}}\right)\\ & =\boxed{ \sqrt{\frac{\pi}{2}} se^{-\frac{s^2}{2}}} \end{align}

Robert Lee
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