Integral of $$$- 9 x^{2} + \frac{81}{x^{3}}$$$

Find $$$\int \left(- 9 x^{2} + \frac{81}{x^{3}}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(- 9 x^{2} + \frac{81}{x^{3}}\right)d x}}} = {\color{red}{\left(\int{\frac{81}{x^{3}} d x} - \int{9 x^{2} d x}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=9$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$$\int{\frac{81}{x^{3}} d x} - {\color{red}{\int{9 x^{2} d x}}} = \int{\frac{81}{x^{3}} d x} - {\color{red}{\left(9 \int{x^{2} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$\int{\frac{81}{x^{3}} d x} - 9 {\color{red}{\int{x^{2} d x}}}=\int{\frac{81}{x^{3}} d x} - 9 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\int{\frac{81}{x^{3}} d x} - 9 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=81$$$ and $$$f{\left(x \right)} = \frac{1}{x^{3}}$$$:

$$- 3 x^{3} + {\color{red}{\int{\frac{81}{x^{3}} d x}}} = - 3 x^{3} + {\color{red}{\left(81 \int{\frac{1}{x^{3}} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-3$$$:

$$- 3 x^{3} + 81 {\color{red}{\int{\frac{1}{x^{3}} d x}}}=- 3 x^{3} + 81 {\color{red}{\int{x^{-3} d x}}}=- 3 x^{3} + 81 {\color{red}{\frac{x^{-3 + 1}}{-3 + 1}}}=- 3 x^{3} + 81 {\color{red}{\left(- \frac{x^{-2}}{2}\right)}}=- 3 x^{3} + 81 {\color{red}{\left(- \frac{1}{2 x^{2}}\right)}}$$

Therefore,

$$\int{\left(- 9 x^{2} + \frac{81}{x^{3}}\right)d x} = - 3 x^{3} - \frac{81}{2 x^{2}}$$

Simplify:

$$\int{\left(- 9 x^{2} + \frac{81}{x^{3}}\right)d x} = \frac{3 \left(- 2 x^{5} - 27\right)}{2 x^{2}}$$

Add the constant of integration:

$$\int{\left(- 9 x^{2} + \frac{81}{x^{3}}\right)d x} = \frac{3 \left(- 2 x^{5} - 27\right)}{2 x^{2}}+C$$

$$$\int \left(- 9 x^{2} + \frac{81}{x^{3}}\right)\, dx = \frac{3 \left(- 2 x^{5} - 27\right)}{2 x^{2}} + C$$$A