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Question Number 128736 by mnjuly1970 last updated on 09/Jan/21

$$...nicecalculus...$$$$$$$$evluate::$$$$\varphi ={\int }_0^{\mathrm{\infty }}{e}^{-x^2}cos\left(x\right)dx=?$$$$$$

Commented by Lordose last updated on 10/Jan/21

Answered by Dwaipayan Shikari last updated on 09/Jan/21

$$\frac{1}{2}{\int }_0^{\mathrm{\infty }}{e}^{-x^2+ix}+e^{-{\mathit{x}}^2-\mathit{i}\mathit{x}}\mathit{d}\mathit{x}$$$$=\frac{1}{2}{\int }_0^{\mathrm{\infty }}{e}^{-{(x-\frac{i}{2})}^2-\frac{1}{4}}+e^{-{(x+\frac{i}{2})}^2-\frac{1}{4}}dx$$$$=\frac{1}{2\sqrt[4]{e}}\left(\sqrt{\pi }\right)=\frac{\sqrt{\pi }}{2\sqrt[4]{e}}$$

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