Integral of $$$- \frac{x^{5}}{6} + 5 x$$$

Find $$$\int \left(- \frac{x^{5}}{6} + 5 x\right)\, dx$$$.

Integrate term by term:

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

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

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

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

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{6}$$$ and $$$f{\left(x \right)} = x^{5}$$$:

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

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

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

Therefore,

$$\int{\left(- \frac{x^{5}}{6} + 5 x\right)d x} = - \frac{x^{6}}{36} + \frac{5 x^{2}}{2}$$

Simplify:

$$\int{\left(- \frac{x^{5}}{6} + 5 x\right)d x} = \frac{x^{2} \left(90 - x^{4}\right)}{36}$$

Add the constant of integration:

$$\int{\left(- \frac{x^{5}}{6} + 5 x\right)d x} = \frac{x^{2} \left(90 - x^{4}\right)}{36}+C$$

$$$\int \left(- \frac{x^{5}}{6} + 5 x\right)\, dx = \frac{x^{2} \left(90 - x^{4}\right)}{36} + C$$$A