Integraal van $$$\frac{e^{- t^{2} x^{2}}}{t^{2}}$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \frac{e^{- t^{2} x^{2}}}{t^{2}}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{1}{t^{2}}$$$ en $$$f{\left(x \right)} = e^{- t^{2} x^{2}}$$$:

$${\color{red}{\int{\frac{e^{- t^{2} x^{2}}}{t^{2}} d x}}} = {\color{red}{\frac{\int{e^{- t^{2} x^{2}} d x}}{t^{2}}}}$$

Zij $$$u=x \left|{t}\right|$$$.

Dan $$$du=\left(x \left|{t}\right|\right)^{\prime }dx = \left|{t}\right| dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = \frac{du}{\left|{t}\right|}$$$.

Dus,

$$\frac{{\color{red}{\int{e^{- t^{2} x^{2}} d x}}}}{t^{2}} = \frac{{\color{red}{\int{\frac{e^{- u^{2}}}{\left|{t}\right|} d u}}}}{t^{2}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{\left|{t}\right|}$$$ en $$$f{\left(u \right)} = e^{- u^{2}}$$$:

$$\frac{{\color{red}{\int{\frac{e^{- u^{2}}}{\left|{t}\right|} d u}}}}{t^{2}} = \frac{{\color{red}{\frac{\int{e^{- u^{2}} d u}}{\left|{t}\right|}}}}{t^{2}}$$

Deze integraal (Foutfunctie) heeft geen gesloten vorm:

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

We herinneren eraan dat $$$u=x \left|{t}\right|$$$:

$$\frac{\sqrt{\pi} \operatorname{erf}{\left({\color{red}{u}} \right)}}{2 t^{2} \left|{t}\right|} = \frac{\sqrt{\pi} \operatorname{erf}{\left({\color{red}{x \left|{t}\right|}} \right)}}{2 t^{2} \left|{t}\right|}$$

Dus,

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

Voeg de integratieconstante toe:

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

$$$\int \frac{e^{- t^{2} x^{2}}}{t^{2}}\, dx = \frac{\sqrt{\pi} \operatorname{erf}{\left(x \left|{t}\right| \right)}}{2 t^{2} \left|{t}\right|} + C$$$A