Integral dari $$$\frac{14}{\left(5 - 3 x\right)^{3}}$$$

Temukan $$$\int \frac{14}{\left(5 - 3 x\right)^{3}}\, dx$$$.

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=14$$$ dan $$$f{\left(x \right)} = \frac{1}{\left(5 - 3 x\right)^{3}}$$$:

$${\color{red}{\int{\frac{14}{\left(5 - 3 x\right)^{3}} d x}}} = {\color{red}{\left(14 \int{\frac{1}{\left(5 - 3 x\right)^{3}} d x}\right)}}$$

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

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

Integralnya menjadi

$$14 {\color{red}{\int{\frac{1}{\left(5 - 3 x\right)^{3}} d x}}} = 14 {\color{red}{\int{\left(- \frac{1}{3 u^{3}}\right)d u}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=- \frac{1}{3}$$$ dan $$$f{\left(u \right)} = \frac{1}{u^{3}}$$$:

$$14 {\color{red}{\int{\left(- \frac{1}{3 u^{3}}\right)d u}}} = 14 {\color{red}{\left(- \frac{\int{\frac{1}{u^{3}} d u}}{3}\right)}}$$

Terapkan aturan pangkat $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ dengan $$$n=-3$$$:

$$- \frac{14 {\color{red}{\int{\frac{1}{u^{3}} d u}}}}{3}=- \frac{14 {\color{red}{\int{u^{-3} d u}}}}{3}=- \frac{14 {\color{red}{\frac{u^{-3 + 1}}{-3 + 1}}}}{3}=- \frac{14 {\color{red}{\left(- \frac{u^{-2}}{2}\right)}}}{3}=- \frac{14 {\color{red}{\left(- \frac{1}{2 u^{2}}\right)}}}{3}$$

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

$$\frac{7 {\color{red}{u}}^{-2}}{3} = \frac{7 {\color{red}{\left(5 - 3 x\right)}}^{-2}}{3}$$

Oleh karena itu,

$$\int{\frac{14}{\left(5 - 3 x\right)^{3}} d x} = \frac{7}{3 \left(5 - 3 x\right)^{2}}$$

Sederhanakan:

$$\int{\frac{14}{\left(5 - 3 x\right)^{3}} d x} = \frac{7}{3 \left(3 x - 5\right)^{2}}$$

Tambahkan konstanta integrasi:

$$\int{\frac{14}{\left(5 - 3 x\right)^{3}} d x} = \frac{7}{3 \left(3 x - 5\right)^{2}}+C$$

$$$\int \frac{14}{\left(5 - 3 x\right)^{3}}\, dx = \frac{7}{3 \left(3 x - 5\right)^{2}} + C$$$A