Integral of $$$- x^{3} + x$$$

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

Integrate term by term:

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

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

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

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

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

Therefore,

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

Simplify:

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

Add the constant of integration:

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

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