Question Number 219970 by Nicholas666 last updated on 04/May/25
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:{e}^{{x}^{\mathrm{2}} } \:{dx} \\ $$$$ \\ $$
Answered by MATHEMATICSAM last updated on 04/May/25
$${e}^{{x}} \:=\:\mathrm{1}\:+\:{x}\:+\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}\:+\:\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}\:+\:\frac{{x}^{\mathrm{4}} }{\mathrm{4}!}\:+\:….\:\infty \\ $$$$\Rightarrow\:{e}^{{x}} \:=\:\underset{{n}\:=\:\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{{x}^{{n}} }{{n}!} \\ $$$${e}^{{x}^{\mathrm{2}} } \:=\:\underset{{n}\:=\:\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{{x}^{\mathrm{2}{n}} }{{n}!} \\ $$$$\Rightarrow\:\int{e}^{{x}^{\mathrm{2}} } \:{dx}\:=\:\underset{{n}\:=\:\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{{x}^{\mathrm{2}{n}\:+\:\mathrm{1}} }{\left(\mathrm{2}{n}\:+\:\mathrm{1}\right){n}!}\:+\:\mathrm{C}\: \\ $$
Answered by MrGaster last updated on 04/May/25
$$\left(\mathrm{1}\right):\int{e}^{{x}^{\mathrm{2}} } =\int\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{x}^{\mathrm{2}{n}} }{{n}!}{dx} \\ $$$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}!}\int{x}^{\mathrm{2}{n}} {dx} \\ $$$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}!}\centerdot\frac{{x}^{\mathrm{2}{n}−\mathrm{1}} }{\mathrm{2}{n}+\mathrm{1}}+{C} \\ $$$$\int{e}^{{x}^{\mathrm{2}} } {dx}=\int\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{x}^{\mathrm{2}{n}+\mathrm{1}} }{{n}!\left(\mathrm{2}{n}+\mathrm{1}\right)}+{C} \\ $$$$\left(\mathrm{2}\right):\int{e}^{{x}^{\mathrm{2}} } =\frac{\sqrt{\pi}}{\mathrm{2}}\mathrm{erfi}\left({x}\right)+{C} \\ $$