Integral of $$$\frac{d}{x} - 6 x - 7$$$ with respect to $$$x$$$

Find $$$\int \left(\frac{d}{x} - 6 x - 7\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(\frac{d}{x} - 6 x - 7\right)d x}}} = {\color{red}{\left(- \int{7 d x} - \int{6 x d x} + \int{\frac{d}{x} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=7$$$:

$$- \int{6 x d x} + \int{\frac{d}{x} d x} - {\color{red}{\int{7 d x}}} = - \int{6 x d x} + \int{\frac{d}{x} d x} - {\color{red}{\left(7 x\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=6$$$ and $$$f{\left(x \right)} = x$$$:

$$- 7 x + \int{\frac{d}{x} d x} - {\color{red}{\int{6 x d x}}} = - 7 x + \int{\frac{d}{x} d x} - {\color{red}{\left(6 \int{x d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=1$$$:

$$- 7 x + \int{\frac{d}{x} d x} - 6 {\color{red}{\int{x d x}}}=- 7 x + \int{\frac{d}{x} d x} - 6 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- 7 x + \int{\frac{d}{x} d x} - 6 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=d$$$ and $$$f{\left(x \right)} = \frac{1}{x}$$$:

$$- 3 x^{2} - 7 x + {\color{red}{\int{\frac{d}{x} d x}}} = - 3 x^{2} - 7 x + {\color{red}{d \int{\frac{1}{x} d x}}}$$

The integral of $$$\frac{1}{x}$$$ is $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

$$d {\color{red}{\int{\frac{1}{x} d x}}} - 3 x^{2} - 7 x = d {\color{red}{\ln{\left(\left|{x}\right| \right)}}} - 3 x^{2} - 7 x$$

Therefore,

$$\int{\left(\frac{d}{x} - 6 x - 7\right)d x} = d \ln{\left(\left|{x}\right| \right)} - 3 x^{2} - 7 x$$

Add the constant of integration:

$$\int{\left(\frac{d}{x} - 6 x - 7\right)d x} = d \ln{\left(\left|{x}\right| \right)} - 3 x^{2} - 7 x+C$$

$$$\int \left(\frac{d}{x} - 6 x - 7\right)\, dx = \left(d \ln\left(\left|{x}\right|\right) - 3 x^{2} - 7 x\right) + C$$$A