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

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

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

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

Jadi,

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

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

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

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

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

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

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

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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