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Question Number 108429 by Rasikh last updated on 16/Aug/20

Answered by Sarah85 last updated on 17/Aug/20

$$\int {\mathrm{e}}^{x^2}dx=\frac{\sqrt{\pi }}{2}\int \frac{{2\mathrm{e}}^{x^2}}{\sqrt{\pi }}dx=\frac{\sqrt{\pi }}{2}\mathrm{erfi}\left(x\right)+C$$

Commented by Rasikh last updated on 17/Aug/20

$$\mathrm{please}\mathrm{solve}\mathrm{step}\mathrm{by}\mathrm{step}$$

Commented by Sarah85 last updated on 17/Aug/20

$$\mathrm{this}\mathrm{is}\mathrm{step}\mathrm{by}\mathrm{step}$$$$\mathrm{erfi}\left(x\right)=\int \frac{{2\mathrm{e}}^{x^2}}{\sqrt{\pi }}dx$$$${\mathrm{it}}^{\prime }\mathrm{s}\mathrm{defined}\mathrm{this}\mathrm{way}$$

Commented by 1549442205PVT last updated on 17/Aug/20

$$\mathrm{As}\mathrm{i}\mathrm{known}\mathrm{erf}\left(\mathrm{x}\right)=\frac{2}{\sqrt{\pi }}{\int }_0^{\mathrm{x}}{\mathrm{e}}^{-{\mathrm{t}}^2}\mathrm{dt}$$$$\mathrm{complementaty}\mathrm{erro}\mathrm{function}\mathrm{is}$$$$\mathrm{defined}\mathrm{as}:{\mathrm{erf}}_{\mathrm{c}}\left(\mathrm{x}\right)={\int }_{\mathrm{x}}^{\mathrm{\infty }}{\mathrm{e}}^{-{\mathrm{t}}^2}\mathrm{dt}$$$$\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{erf}\left(\mathrm{x}\right)\right)=\frac{2}{\sqrt{\pi }}{\mathrm{e}}^{-{\mathrm{x}}^2},\mathrm{so}\mathrm{set}{\mathrm{x}}^2=-{\mathrm{u}}^2={\mathrm{i}}^2{\mathrm{u}}^2$$

Commented by Sarah85 last updated on 17/Aug/20

$$\mathrm{why},\mathrm{when}\mathrm{we}\mathrm{also}\mathrm{have}\mathrm{erfi}\left(x\right)?$$

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