Integral of $$$\frac{310}{x^{32}}$$$

Find $$$\int \frac{310}{x^{32}}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=310$$$ and $$$f{\left(x \right)} = \frac{1}{x^{32}}$$$:

$${\color{red}{\int{\frac{310}{x^{32}} d x}}} = {\color{red}{\left(310 \int{\frac{1}{x^{32}} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-32$$$:

$$310 {\color{red}{\int{\frac{1}{x^{32}} d x}}}=310 {\color{red}{\int{x^{-32} d x}}}=310 {\color{red}{\frac{x^{-32 + 1}}{-32 + 1}}}=310 {\color{red}{\left(- \frac{x^{-31}}{31}\right)}}=310 {\color{red}{\left(- \frac{1}{31 x^{31}}\right)}}$$

Therefore,

$$\int{\frac{310}{x^{32}} d x} = - \frac{10}{x^{31}}$$

Add the constant of integration:

$$\int{\frac{310}{x^{32}} d x} = - \frac{10}{x^{31}}+C$$

$$$\int \frac{310}{x^{32}}\, dx = - \frac{10}{x^{31}} + C$$$A