0-e-x-2-dx-

Question Number 202842 by esmaeil last updated on 04/Jan/24

$$\int_{\mathrm{0}} ^{\infty} {e}^{−{x}^{\mathrm{2}} } {dx} \\ $$

Commented by esmaeil last updated on 04/Jan/24

$$\:\cancel{\underline{\mathcal{W}}}{this} \\ $$$$\:\cancel{\underbrace{\mathcal{W}}} \\ $$$$ \\ $$

Answered by MathematicalUser2357 last updated on 04/Jan/24

$$=\frac{\sqrt{\pi}}{\mathrm{2}}\approx\mathrm{0}.\mathrm{886227} \\ $$

Commented by esmaeil last updated on 04/Jan/24

$${tnanks}\:\:\left({solution}?\right) \\ $$

Commented by MathematicalUser2357 last updated on 04/Jan/24

$$\int{e}^{−{x}^{\mathrm{2}} } {dx}=\frac{\sqrt{\pi}}{\mathrm{2}}\mathrm{erf}\left({x}\right)+{C} \\ $$$$\int_{\mathrm{0}} ^{\infty} {e}^{−{x}^{\mathrm{2}} } {dx}=\left[\frac{\sqrt{\pi}}{\mathrm{2}}\mathrm{erf}\left({x}\right)\right]_{\mathrm{0}} ^{\infty} =\frac{\sqrt{\pi}}{\mathrm{2}} \\ $$

Answered by aleks041103 last updated on 04/Jan/24

$$\int_{\mathrm{0}} ^{\:\infty} {e}^{−{x}^{\mathrm{2}} } {dx}=\frac{\mathrm{1}}{\mathrm{2}}\int_{−\infty} ^{\:\infty} {e}^{−{x}^{\mathrm{2}} } {dx} \\ $$$${I}=\int_{−\infty} ^{\:\infty} {e}^{−{x}^{\mathrm{2}} } {dx}=\int_{\mathbb{R}} {e}^{−{x}^{\mathrm{2}} } {dx} \\ $$$${x}\leftrightarrow{y} \\ $$$${I}=\int_{\mathbb{R}} {e}^{−{y}^{\mathrm{2}} } {dy} \\ $$$$\Rightarrow{I}^{\mathrm{2}} =\left(\int_{\mathbb{R}} {e}^{−{x}^{\mathrm{2}} } {dx}\right)\left(\int_{\mathbb{R}} {e}^{−{y}^{\mathrm{2}} } {dy}\right)= \\ $$$$=\int_{\mathbb{R}} \int_{\mathbb{R}} {e}^{−{x}^{\mathrm{2}} } {e}^{−{y}^{\mathrm{2}} } {dxdy}= \\ $$$$=\int\int_{\mathbb{R}^{\mathrm{2}} } {e}^{−\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)} {dxdy} \\ $$$${Change}\:{to}\:{polar}\:{coord} \\ $$$${r}^{\mathrm{2}} ={x}^{\mathrm{2}} +{y}^{\mathrm{2}} \\ $$$${dxdy}={rdrd}\theta \\ $$$$\Rightarrow{I}^{\mathrm{2}} =\int\int_{\mathbb{R}^{\mathrm{2}} } {e}^{−\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)} {dxdy}= \\ $$$$=\int_{\mathrm{0}} ^{\:\infty} \left(\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} {re}^{−{r}^{\mathrm{2}} } {d}\theta\right){dr}= \\ $$$$=\left(\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} {d}\theta\right)\left(\int_{\mathrm{0}} ^{\:\infty} {re}^{−{r}^{\mathrm{2}} } {dr}\right)= \\ $$$$=\mathrm{2}\pi\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\infty} {e}^{−{r}^{\mathrm{2}} } {d}\left({r}^{\mathrm{2}} \right)= \\ $$$$=\pi\left({e}^{−\mathrm{0}^{\mathrm{2}} } −\underset{{r}\rightarrow\infty} {{lim}}\:{e}^{−{r}^{\mathrm{2}} } \right)=\pi \\ $$$$\Rightarrow{I}=\sqrt{\pi} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} {e}^{−{x}^{\mathrm{2}} } {dx}=\frac{{I}}{\mathrm{2}} \\ $$$$\Rightarrow\int_{\mathrm{0}} ^{\:\infty} {e}^{−{x}^{\mathrm{2}} } {dx}=\frac{\sqrt{\pi}}{\mathrm{2}} \\ $$

Commented by esmaeil last updated on 04/Jan/24

$${thanks} \\ $$

Answered by Mathspace last updated on 05/Jan/24

$${I}=\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}^{\mathrm{2}} } {dx}=_{{x}=\sqrt{{t}}} \:\int_{\mathrm{0}} ^{\infty} {e}^{−{t}} \frac{{dt}}{\mathrm{2}\sqrt{{t}}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} {t}^{−\frac{\mathrm{1}}{\mathrm{2}}} {e}^{−{t}} {dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} {t}^{\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{1}} {e}^{−{t}} {dt}=\frac{\mathrm{1}}{\mathrm{2}}\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\pi}=\frac{\sqrt{\pi}}{\mathrm{2}} \\ $$

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