Integral dari $$$\frac{1}{\left(3 - 5 x\right)^{2}}$$$

Temukan $$$\int \frac{1}{\left(3 - 5 x\right)^{2}}\, dx$$$.

Misalkan $$$u=3 - 5 x$$$.

Kemudian $$$du=\left(3 - 5 x\right)^{\prime }dx = - 5 dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dx = - \frac{du}{5}$$$.

Jadi,

$${\color{red}{\int{\frac{1}{\left(3 - 5 x\right)^{2}} d x}}} = {\color{red}{\int{\left(- \frac{1}{5 u^{2}}\right)d u}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=- \frac{1}{5}$$$ dan $$$f{\left(u \right)} = \frac{1}{u^{2}}$$$:

$${\color{red}{\int{\left(- \frac{1}{5 u^{2}}\right)d u}}} = {\color{red}{\left(- \frac{\int{\frac{1}{u^{2}} d u}}{5}\right)}}$$

Terapkan aturan pangkat $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ dengan $$$n=-2$$$:

$$- \frac{{\color{red}{\int{\frac{1}{u^{2}} d u}}}}{5}=- \frac{{\color{red}{\int{u^{-2} d u}}}}{5}=- \frac{{\color{red}{\frac{u^{-2 + 1}}{-2 + 1}}}}{5}=- \frac{{\color{red}{\left(- u^{-1}\right)}}}{5}=- \frac{{\color{red}{\left(- \frac{1}{u}\right)}}}{5}$$

Ingat bahwa $$$u=3 - 5 x$$$:

$$\frac{{\color{red}{u}}^{-1}}{5} = \frac{{\color{red}{\left(3 - 5 x\right)}}^{-1}}{5}$$

Oleh karena itu,

$$\int{\frac{1}{\left(3 - 5 x\right)^{2}} d x} = \frac{1}{5 \left(3 - 5 x\right)}$$

Sederhanakan:

$$\int{\frac{1}{\left(3 - 5 x\right)^{2}} d x} = - \frac{1}{25 x - 15}$$

Tambahkan konstanta integrasi:

$$\int{\frac{1}{\left(3 - 5 x\right)^{2}} d x} = - \frac{1}{25 x - 15}+C$$

$$$\int \frac{1}{\left(3 - 5 x\right)^{2}}\, dx = - \frac{1}{25 x - 15} + C$$$A