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

Find $$$\int s x^{- m} x^{n}\, dx$$$.

The input is rewritten: $$$\int{s x^{- m} x^{n} d x}=\int{s x^{- m + n} d x}$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=s$$$ and $$$f{\left(x \right)} = x^{- m + n}$$$:

$${\color{red}{\int{s x^{- m + n} d x}}} = {\color{red}{s \int{x^{- m + n} d x}}}$$

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

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

Therefore,

$$\int{s x^{- m + n} d x} = \frac{s x^{- m + n + 1}}{- m + n + 1}$$

Add the constant of integration:

$$\int{s x^{- m + n} d x} = \frac{s x^{- m + n + 1}}{- m + n + 1}+C$$

$$$\int s x^{- m} x^{n}\, dx = \frac{s x^{- m + n + 1}}{- m + n + 1} + C$$$A