Integral dari $$$\frac{\sqrt{11} e^{- \frac{x}{2}}}{22}$$$

Temukan $$$\int \frac{\sqrt{11} e^{- \frac{x}{2}}}{22}\, dx$$$.

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=\frac{\sqrt{11}}{22}$$$ dan $$$f{\left(x \right)} = e^{- \frac{x}{2}}$$$:

$${\color{red}{\int{\frac{\sqrt{11} e^{- \frac{x}{2}}}{22} d x}}} = {\color{red}{\left(\frac{\sqrt{11} \int{e^{- \frac{x}{2}} d x}}{22}\right)}}$$

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

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

Integralnya menjadi

$$\frac{\sqrt{11} {\color{red}{\int{e^{- \frac{x}{2}} d x}}}}{22} = \frac{\sqrt{11} {\color{red}{\int{\left(- 2 e^{u}\right)d u}}}}{22}$$

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

$$\frac{\sqrt{11} {\color{red}{\int{\left(- 2 e^{u}\right)d u}}}}{22} = \frac{\sqrt{11} {\color{red}{\left(- 2 \int{e^{u} d u}\right)}}}{22}$$

Integral dari fungsi eksponensial adalah $$$\int{e^{u} d u} = e^{u}$$$:

$$- \frac{\sqrt{11} {\color{red}{\int{e^{u} d u}}}}{11} = - \frac{\sqrt{11} {\color{red}{e^{u}}}}{11}$$

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

$$- \frac{\sqrt{11} e^{{\color{red}{u}}}}{11} = - \frac{\sqrt{11} e^{{\color{red}{\left(- \frac{x}{2}\right)}}}}{11}$$

Oleh karena itu,

$$\int{\frac{\sqrt{11} e^{- \frac{x}{2}}}{22} d x} = - \frac{\sqrt{11} e^{- \frac{x}{2}}}{11}$$

Tambahkan konstanta integrasi:

$$\int{\frac{\sqrt{11} e^{- \frac{x}{2}}}{22} d x} = - \frac{\sqrt{11} e^{- \frac{x}{2}}}{11}+C$$

$$$\int \frac{\sqrt{11} e^{- \frac{x}{2}}}{22}\, dx = - \frac{\sqrt{11} e^{- \frac{x}{2}}}{11} + C$$$A