Integral of $$$x^{\frac{21}{10}} - x^{2}$$$

Find $$$\int \left(x^{\frac{21}{10}} - x^{2}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(x^{\frac{21}{10}} - x^{2}\right)d x}}} = {\color{red}{\left(- \int{x^{2} d x} + \int{x^{\frac{21}{10}} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=\frac{21}{10}$$$:

$$- \int{x^{2} d x} + {\color{red}{\int{x^{\frac{21}{10}} d x}}}=- \int{x^{2} d x} + {\color{red}{\frac{x^{1 + \frac{21}{10}}}{1 + \frac{21}{10}}}}=- \int{x^{2} d x} + {\color{red}{\left(\frac{10 x^{\frac{31}{10}}}{31}\right)}}$$

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

$$\frac{10 x^{\frac{31}{10}}}{31} - {\color{red}{\int{x^{2} d x}}}=\frac{10 x^{\frac{31}{10}}}{31} - {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{10 x^{\frac{31}{10}}}{31} - {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Therefore,

$$\int{\left(x^{\frac{21}{10}} - x^{2}\right)d x} = \frac{10 x^{\frac{31}{10}}}{31} - \frac{x^{3}}{3}$$

Add the constant of integration:

$$\int{\left(x^{\frac{21}{10}} - x^{2}\right)d x} = \frac{10 x^{\frac{31}{10}}}{31} - \frac{x^{3}}{3}+C$$

$$$\int \left(x^{\frac{21}{10}} - x^{2}\right)\, dx = \left(\frac{10 x^{\frac{31}{10}}}{31} - \frac{x^{3}}{3}\right) + C$$$A