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Question Number 196412 by Erico last updated on 25/Aug/23

$$\mathrm{Prove}\mathrm{that}{\underset{i}{\int }}^{+\mathrm{\infty }}(\cos 2\mathrm{t}+\mathrm{isin}2\mathrm{t}){\mathrm{e}}^{-{\mathrm{t}}^2}\mathrm{dt}=\frac{\sqrt{\pi }}{2\mathrm{e}}$$

Commented by JDamian last updated on 24/Aug/23

$$is\mathbf{t}realorcomplex?$$

Answered by MrGaster last updated on 16/Feb/25

$${\int }_0^{+\mathrm{\infty }}\cos 2t\cdot e^{-t^2}dt=\frac{\sqrt{\pi }}{2e}$$$${\int }_0^{+\mathrm{\infty }}\sin 2t\cdot e^{-t^2}dt=0$$$${\underset{i}{\int }}^{+\mathrm{\infty }}(\cos 2\mathrm{t}+\mathrm{isin}2\mathrm{t}){\mathrm{e}}^{-{\mathrm{t}}^2}\mathrm{dt}=\frac{\sqrt{\pi }}{2\mathrm{e}}$$

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