Integralen av $$$e^{- \frac{y^{2}}{2}}$$$

Bestäm $$$\int e^{- \frac{y^{2}}{2}}\, dy$$$.

Låt $$$u=\frac{\sqrt{2} y}{2}$$$ vara.

Då $$$du=\left(\frac{\sqrt{2} y}{2}\right)^{\prime }dy = \frac{\sqrt{2}}{2} dy$$$ (stegen kan ses »), och vi har att $$$dy = \sqrt{2} du$$$.

Integralen kan omskrivas som

$${\color{red}{\int{e^{- \frac{y^{2}}{2}} d y}}} = {\color{red}{\int{\sqrt{2} e^{- u^{2}} d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\sqrt{2}$$$ och $$$f{\left(u \right)} = e^{- u^{2}}$$$:

$${\color{red}{\int{\sqrt{2} e^{- u^{2}} d u}}} = {\color{red}{\sqrt{2} \int{e^{- u^{2}} d u}}}$$

Denna integral (Felintegral) har ingen sluten form:

$$\sqrt{2} {\color{red}{\int{e^{- u^{2}} d u}}} = \sqrt{2} {\color{red}{\left(\frac{\sqrt{\pi} \operatorname{erf}{\left(u \right)}}{2}\right)}}$$

Kom ihåg att $$$u=\frac{\sqrt{2} y}{2}$$$:

$$\frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left({\color{red}{u}} \right)}}{2} = \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left({\color{red}{\left(\frac{\sqrt{2} y}{2}\right)}} \right)}}{2}$$

Alltså,

$$\int{e^{- \frac{y^{2}}{2}} d y} = \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} y}{2} \right)}}{2}$$

Lägg till integrationskonstanten:

$$\int{e^{- \frac{y^{2}}{2}} d y} = \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} y}{2} \right)}}{2}+C$$

$$$\int e^{- \frac{y^{2}}{2}}\, dy = \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} y}{2} \right)}}{2} + C$$$A