Integral of $$$\frac{3}{x^{25}}$$$

Find $$$\int \frac{3}{x^{25}}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=3$$$ and $$$f{\left(x \right)} = \frac{1}{x^{25}}$$$:

$${\color{red}{\int{\frac{3}{x^{25}} d x}}} = {\color{red}{\left(3 \int{\frac{1}{x^{25}} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-25$$$:

$$3 {\color{red}{\int{\frac{1}{x^{25}} d x}}}=3 {\color{red}{\int{x^{-25} d x}}}=3 {\color{red}{\frac{x^{-25 + 1}}{-25 + 1}}}=3 {\color{red}{\left(- \frac{x^{-24}}{24}\right)}}=3 {\color{red}{\left(- \frac{1}{24 x^{24}}\right)}}$$

Therefore,

$$\int{\frac{3}{x^{25}} d x} = - \frac{1}{8 x^{24}}$$

Add the constant of integration:

$$\int{\frac{3}{x^{25}} d x} = - \frac{1}{8 x^{24}}+C$$

$$$\int \frac{3}{x^{25}}\, dx = - \frac{1}{8 x^{24}} + C$$$A