Integral dari $$$\frac{\sqrt{2} e^{- \frac{x^{2}}{2}}}{2 \sqrt{\pi}}$$$

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

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=\frac{\sqrt{2}}{2 \sqrt{\pi}}$$$ dan $$$f{\left(x \right)} = e^{- \frac{x^{2}}{2}}$$$:

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

Misalkan $$$u=\frac{\sqrt{2} x}{2}$$$.

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

Oleh karena itu,

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

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

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

Integral ini (Fungsi galat) tidak memiliki bentuk tertutup:

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

Ingat bahwa $$$u=\frac{\sqrt{2} x}{2}$$$:

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

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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