Integral of $$$x^{- \alpha}$$$ with respect to $$$x$$$

Find $$$\int x^{- \alpha}\, dx$$$.

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

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

Therefore,

$$\int{x^{- \alpha} d x} = \frac{x^{1 - \alpha}}{1 - \alpha}$$

Simplify:

$$\int{x^{- \alpha} d x} = - \frac{x^{1 - \alpha}}{\alpha - 1}$$

Add the constant of integration:

$$\int{x^{- \alpha} d x} = - \frac{x^{1 - \alpha}}{\alpha - 1}+C$$

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