Integral of $$$- 3 x^{5} - 10 x^{2} - 4 x$$$

Find $$$\int \left(- 3 x^{5} - 10 x^{2} - 4 x\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(- 3 x^{5} - 10 x^{2} - 4 x\right)d x}}} = {\color{red}{\left(- \int{4 x d x} - \int{10 x^{2} d x} - \int{3 x^{5} d x}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=10$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$$- \int{4 x d x} - \int{3 x^{5} d x} - {\color{red}{\int{10 x^{2} d x}}} = - \int{4 x d x} - \int{3 x^{5} d x} - {\color{red}{\left(10 \int{x^{2} d x}\right)}}$$

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

$$- \int{4 x d x} - \int{3 x^{5} d x} - 10 {\color{red}{\int{x^{2} d x}}}=- \int{4 x d x} - \int{3 x^{5} d x} - 10 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \int{4 x d x} - \int{3 x^{5} d x} - 10 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=4$$$ and $$$f{\left(x \right)} = x$$$:

$$- \frac{10 x^{3}}{3} - \int{3 x^{5} d x} - {\color{red}{\int{4 x d x}}} = - \frac{10 x^{3}}{3} - \int{3 x^{5} d x} - {\color{red}{\left(4 \int{x d x}\right)}}$$

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

$$- \frac{10 x^{3}}{3} - \int{3 x^{5} d x} - 4 {\color{red}{\int{x d x}}}=- \frac{10 x^{3}}{3} - \int{3 x^{5} d x} - 4 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- \frac{10 x^{3}}{3} - \int{3 x^{5} d x} - 4 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=3$$$ and $$$f{\left(x \right)} = x^{5}$$$:

$$- \frac{10 x^{3}}{3} - 2 x^{2} - {\color{red}{\int{3 x^{5} d x}}} = - \frac{10 x^{3}}{3} - 2 x^{2} - {\color{red}{\left(3 \int{x^{5} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=5$$$:

$$- \frac{10 x^{3}}{3} - 2 x^{2} - 3 {\color{red}{\int{x^{5} d x}}}=- \frac{10 x^{3}}{3} - 2 x^{2} - 3 {\color{red}{\frac{x^{1 + 5}}{1 + 5}}}=- \frac{10 x^{3}}{3} - 2 x^{2} - 3 {\color{red}{\left(\frac{x^{6}}{6}\right)}}$$

Therefore,

$$\int{\left(- 3 x^{5} - 10 x^{2} - 4 x\right)d x} = - \frac{x^{6}}{2} - \frac{10 x^{3}}{3} - 2 x^{2}$$

Simplify:

$$\int{\left(- 3 x^{5} - 10 x^{2} - 4 x\right)d x} = \frac{x^{2} \left(- 3 x^{4} - 20 x - 12\right)}{6}$$

Add the constant of integration:

$$\int{\left(- 3 x^{5} - 10 x^{2} - 4 x\right)d x} = \frac{x^{2} \left(- 3 x^{4} - 20 x - 12\right)}{6}+C$$

$$$\int \left(- 3 x^{5} - 10 x^{2} - 4 x\right)\, dx = \frac{x^{2} \left(- 3 x^{4} - 20 x - 12\right)}{6} + C$$$A