Integral of $$$- x^{6} - x^{3} - 111 i x^{3}$$$

Find $$$\int \left(- x^{6} - x^{3} - 111 i x^{3}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(- x^{6} - x^{3} - 111 i x^{3}\right)d x}}} = {\color{red}{\left(- \int{x^{3} d x} - \int{x^{6} d x} - \int{111 i x^{3} d x}\right)}}$$

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

$$- \int{x^{6} d x} - \int{111 i x^{3} d x} - {\color{red}{\int{x^{3} d x}}}=- \int{x^{6} d x} - \int{111 i x^{3} d x} - {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- \int{x^{6} d x} - \int{111 i x^{3} d x} - {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=6$$$:

$$- \frac{x^{4}}{4} - \int{111 i x^{3} d x} - {\color{red}{\int{x^{6} d x}}}=- \frac{x^{4}}{4} - \int{111 i x^{3} d x} - {\color{red}{\frac{x^{1 + 6}}{1 + 6}}}=- \frac{x^{4}}{4} - \int{111 i x^{3} d x} - {\color{red}{\left(\frac{x^{7}}{7}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=111 i$$$ and $$$f{\left(x \right)} = x^{3}$$$:

$$- \frac{x^{7}}{7} - \frac{x^{4}}{4} - {\color{red}{\int{111 i x^{3} d x}}} = - \frac{x^{7}}{7} - \frac{x^{4}}{4} - {\color{red}{\left(111 i \int{x^{3} d x}\right)}}$$

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

$$- \frac{x^{7}}{7} - \frac{x^{4}}{4} - 111 i {\color{red}{\int{x^{3} d x}}}=- \frac{x^{7}}{7} - \frac{x^{4}}{4} - 111 i {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- \frac{x^{7}}{7} - \frac{x^{4}}{4} - 111 i {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Therefore,

$$\int{\left(- x^{6} - x^{3} - 111 i x^{3}\right)d x} = - \frac{x^{7}}{7} - \frac{x^{4}}{4} - \frac{111 i x^{4}}{4}$$

Simplify:

$$\int{\left(- x^{6} - x^{3} - 111 i x^{3}\right)d x} = \frac{x^{4} \left(- 4 x^{3} - 7 - 777 i\right)}{28}$$

Add the constant of integration:

$$\int{\left(- x^{6} - x^{3} - 111 i x^{3}\right)d x} = \frac{x^{4} \left(- 4 x^{3} - 7 - 777 i\right)}{28}+C$$

$$$\int \left(- x^{6} - x^{3} - 111 i x^{3}\right)\, dx = \frac{x^{4} \left(- 4 x^{3} - 7 - 777 i\right)}{28} + C$$$A