Integral dari $$$e^{\frac{x}{c}}$$$ terhadap $$$x$$$

Temukan $$$\int e^{\frac{x}{c}}\, dx$$$.

Misalkan $$$u=\frac{x}{c}$$$.

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

Jadi,

$${\color{red}{\int{e^{\frac{x}{c}} d x}}} = {\color{red}{\int{c e^{u} d u}}}$$

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

$${\color{red}{\int{c e^{u} d u}}} = {\color{red}{c \int{e^{u} d u}}}$$

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

$$c {\color{red}{\int{e^{u} d u}}} = c {\color{red}{e^{u}}}$$

Ingat bahwa $$$u=\frac{x}{c}$$$:

$$c e^{{\color{red}{u}}} = c e^{{\color{red}{\frac{x}{c}}}}$$

Oleh karena itu,

$$\int{e^{\frac{x}{c}} d x} = c e^{\frac{x}{c}}$$

Tambahkan konstanta integrasi:

$$\int{e^{\frac{x}{c}} d x} = c e^{\frac{x}{c}}+C$$

$$$\int e^{\frac{x}{c}}\, dx = c e^{\frac{x}{c}} + C$$$A