Integral of $$$- 2 x^{5} + 13 x^{2}$$$

Find $$$\int \left(- 2 x^{5} + 13 x^{2}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(- 2 x^{5} + 13 x^{2}\right)d x}}} = {\color{red}{\left(\int{13 x^{2} d x} - \int{2 x^{5} d x}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = x^{5}$$$:

$$\int{13 x^{2} d x} - {\color{red}{\int{2 x^{5} d x}}} = \int{13 x^{2} d x} - {\color{red}{\left(2 \int{x^{5} d x}\right)}}$$

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

$$\int{13 x^{2} d x} - 2 {\color{red}{\int{x^{5} d x}}}=\int{13 x^{2} d x} - 2 {\color{red}{\frac{x^{1 + 5}}{1 + 5}}}=\int{13 x^{2} d x} - 2 {\color{red}{\left(\frac{x^{6}}{6}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=13$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$$- \frac{x^{6}}{3} + {\color{red}{\int{13 x^{2} d x}}} = - \frac{x^{6}}{3} + {\color{red}{\left(13 \int{x^{2} d x}\right)}}$$

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

$$- \frac{x^{6}}{3} + 13 {\color{red}{\int{x^{2} d x}}}=- \frac{x^{6}}{3} + 13 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \frac{x^{6}}{3} + 13 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Therefore,

$$\int{\left(- 2 x^{5} + 13 x^{2}\right)d x} = - \frac{x^{6}}{3} + \frac{13 x^{3}}{3}$$

Simplify:

$$\int{\left(- 2 x^{5} + 13 x^{2}\right)d x} = \frac{x^{3} \left(13 - x^{3}\right)}{3}$$

Add the constant of integration:

$$\int{\left(- 2 x^{5} + 13 x^{2}\right)d x} = \frac{x^{3} \left(13 - x^{3}\right)}{3}+C$$

$$$\int \left(- 2 x^{5} + 13 x^{2}\right)\, dx = \frac{x^{3} \left(13 - x^{3}\right)}{3} + C$$$A