Integral dari $$$e^{32 y}$$$

Temukan $$$\int e^{32 y}\, dy$$$.

Misalkan $$$u=32 y$$$.

Kemudian $$$du=\left(32 y\right)^{\prime }dy = 32 dy$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dy = \frac{du}{32}$$$.

Jadi,

$${\color{red}{\int{e^{32 y} d y}}} = {\color{red}{\int{\frac{e^{u}}{32} d u}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=\frac{1}{32}$$$ dan $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{\frac{e^{u}}{32} d u}}} = {\color{red}{\left(\frac{\int{e^{u} d u}}{32}\right)}}$$

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

$$\frac{{\color{red}{\int{e^{u} d u}}}}{32} = \frac{{\color{red}{e^{u}}}}{32}$$

Ingat bahwa $$$u=32 y$$$:

$$\frac{e^{{\color{red}{u}}}}{32} = \frac{e^{{\color{red}{\left(32 y\right)}}}}{32}$$

Oleh karena itu,

$$\int{e^{32 y} d y} = \frac{e^{32 y}}{32}$$

Tambahkan konstanta integrasi:

$$\int{e^{32 y} d y} = \frac{e^{32 y}}{32}+C$$

$$$\int e^{32 y}\, dy = \frac{e^{32 y}}{32} + C$$$A