Integral of $$$- 19 x^{5} + x^{4} + 2 x^{3}$$$

Find $$$\int \left(- 19 x^{5} + x^{4} + 2 x^{3}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(- 19 x^{5} + x^{4} + 2 x^{3}\right)d x}}} = {\color{red}{\left(\int{2 x^{3} d x} + \int{x^{4} d x} - \int{19 x^{5} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=4$$$:

$$\int{2 x^{3} d x} - \int{19 x^{5} d x} + {\color{red}{\int{x^{4} d x}}}=\int{2 x^{3} d x} - \int{19 x^{5} d x} + {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=\int{2 x^{3} d x} - \int{19 x^{5} d x} + {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=19$$$ and $$$f{\left(x \right)} = x^{5}$$$:

$$\frac{x^{5}}{5} + \int{2 x^{3} d x} - {\color{red}{\int{19 x^{5} d x}}} = \frac{x^{5}}{5} + \int{2 x^{3} d x} - {\color{red}{\left(19 \int{x^{5} d x}\right)}}$$

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

$$\frac{x^{5}}{5} + \int{2 x^{3} d x} - 19 {\color{red}{\int{x^{5} d x}}}=\frac{x^{5}}{5} + \int{2 x^{3} d x} - 19 {\color{red}{\frac{x^{1 + 5}}{1 + 5}}}=\frac{x^{5}}{5} + \int{2 x^{3} d x} - 19 {\color{red}{\left(\frac{x^{6}}{6}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = x^{3}$$$:

$$- \frac{19 x^{6}}{6} + \frac{x^{5}}{5} + {\color{red}{\int{2 x^{3} d x}}} = - \frac{19 x^{6}}{6} + \frac{x^{5}}{5} + {\color{red}{\left(2 \int{x^{3} d x}\right)}}$$

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

$$- \frac{19 x^{6}}{6} + \frac{x^{5}}{5} + 2 {\color{red}{\int{x^{3} d x}}}=- \frac{19 x^{6}}{6} + \frac{x^{5}}{5} + 2 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- \frac{19 x^{6}}{6} + \frac{x^{5}}{5} + 2 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Therefore,

$$\int{\left(- 19 x^{5} + x^{4} + 2 x^{3}\right)d x} = - \frac{19 x^{6}}{6} + \frac{x^{5}}{5} + \frac{x^{4}}{2}$$

Simplify:

$$\int{\left(- 19 x^{5} + x^{4} + 2 x^{3}\right)d x} = \frac{x^{4} \left(- 95 x^{2} + 6 x + 15\right)}{30}$$

Add the constant of integration:

$$\int{\left(- 19 x^{5} + x^{4} + 2 x^{3}\right)d x} = \frac{x^{4} \left(- 95 x^{2} + 6 x + 15\right)}{30}+C$$

$$$\int \left(- 19 x^{5} + x^{4} + 2 x^{3}\right)\, dx = \frac{x^{4} \left(- 95 x^{2} + 6 x + 15\right)}{30} + C$$$A