Integral dari $$$e^{- \frac{6 x}{5}}$$$

Temukan $$$\int e^{- \frac{6 x}{5}}\, dx$$$.

Misalkan $$$u=- \frac{6 x}{5}$$$.

Kemudian $$$du=\left(- \frac{6 x}{5}\right)^{\prime }dx = - \frac{6 dx}{5}$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dx = - \frac{5 du}{6}$$$.

Jadi,

$${\color{red}{\int{e^{- \frac{6 x}{5}} d x}}} = {\color{red}{\int{\left(- \frac{5 e^{u}}{6}\right)d u}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=- \frac{5}{6}$$$ dan $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{\left(- \frac{5 e^{u}}{6}\right)d u}}} = {\color{red}{\left(- \frac{5 \int{e^{u} d u}}{6}\right)}}$$

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

$$- \frac{5 {\color{red}{\int{e^{u} d u}}}}{6} = - \frac{5 {\color{red}{e^{u}}}}{6}$$

Ingat bahwa $$$u=- \frac{6 x}{5}$$$:

$$- \frac{5 e^{{\color{red}{u}}}}{6} = - \frac{5 e^{{\color{red}{\left(- \frac{6 x}{5}\right)}}}}{6}$$

Oleh karena itu,

$$\int{e^{- \frac{6 x}{5}} d x} = - \frac{5 e^{- \frac{6 x}{5}}}{6}$$

Tambahkan konstanta integrasi:

$$\int{e^{- \frac{6 x}{5}} d x} = - \frac{5 e^{- \frac{6 x}{5}}}{6}+C$$

$$$\int e^{- \frac{6 x}{5}}\, dx = - \frac{5 e^{- \frac{6 x}{5}}}{6} + C$$$A