Integral dari $$$-25 + \frac{5}{x}$$$

Temukan $$$\int \left(-25 + \frac{5}{x}\right)\, dx$$$.

Integralkan suku demi suku:

$${\color{red}{\int{\left(-25 + \frac{5}{x}\right)d x}}} = {\color{red}{\left(- \int{25 d x} + \int{\frac{5}{x} d x}\right)}}$$

Terapkan aturan konstanta $$$\int c\, dx = c x$$$ dengan $$$c=25$$$:

$$\int{\frac{5}{x} d x} - {\color{red}{\int{25 d x}}} = \int{\frac{5}{x} d x} - {\color{red}{\left(25 x\right)}}$$

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

$$- 25 x + {\color{red}{\int{\frac{5}{x} d x}}} = - 25 x + {\color{red}{\left(5 \int{\frac{1}{x} d x}\right)}}$$

Integral dari $$$\frac{1}{x}$$$ adalah $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

$$- 25 x + 5 {\color{red}{\int{\frac{1}{x} d x}}} = - 25 x + 5 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

Oleh karena itu,

$$\int{\left(-25 + \frac{5}{x}\right)d x} = - 25 x + 5 \ln{\left(\left|{x}\right| \right)}$$

Tambahkan konstanta integrasi:

$$\int{\left(-25 + \frac{5}{x}\right)d x} = - 25 x + 5 \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \left(-25 + \frac{5}{x}\right)\, dx = \left(- 25 x + 5 \ln\left(\left|{x}\right|\right)\right) + C$$$A