Integral dari $$$e^{- \frac{x^{2}}{2 \sigma^{2}}}$$$ terhadap $$$x$$$

Temukan $$$\int e^{- \frac{x^{2}}{2 \sigma^{2}}}\, dx$$$.

Misalkan $$$u=\frac{\sqrt{2} x}{2 \left|{\sigma}\right|}$$$.

Kemudian $$$du=\left(\frac{\sqrt{2} x}{2 \left|{\sigma}\right|}\right)^{\prime }dx = \frac{\sqrt{2}}{2 \left|{\sigma}\right|} dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dx = \sqrt{2} \left|{\sigma}\right| du$$$.

Dengan demikian,

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

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=\sqrt{2} \left|{\sigma}\right|$$$ dan $$$f{\left(u \right)} = e^{- u^{2}}$$$:

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

Integral ini (Fungsi galat) tidak memiliki bentuk tertutup:

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

Ingat bahwa $$$u=\frac{\sqrt{2} x}{2 \left|{\sigma}\right|}$$$:

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

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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