Integral of $$$x - x x^{- s} - 1$$$ with respect to $$$x$$$

Find $$$\int \left(x - x x^{- s} - 1\right)\, dx$$$.

The input is rewritten: $$$\int{\left(x - x x^{- s} - 1\right)d x}=\int{\left(x - x^{1 - s} - 1\right)d x}$$$.

Integrate term by term:

$${\color{red}{\int{\left(x - x^{1 - s} - 1\right)d x}}} = {\color{red}{\left(- \int{1 d x} + \int{x d x} - \int{x^{1 - s} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=1$$$:

$$\int{x d x} - \int{x^{1 - s} d x} - {\color{red}{\int{1 d x}}} = \int{x d x} - \int{x^{1 - s} d x} - {\color{red}{x}}$$

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

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

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

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

Therefore,

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

Simplify:

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

Add the constant of integration:

$$\int{\left(x - x^{1 - s} - 1\right)d x} = \frac{\frac{x \left(s - 2\right) \left(x - 2\right)}{2} + x^{2 - s}}{s - 2}+C$$

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