Integral de $$$- 3 y^{5} + y^{2}$$$

Encontre $$$\int \left(- 3 y^{5} + y^{2}\right)\, dy$$$.

Integre termo a termo:

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

Aplique a regra da potência $$$\int y^{n}\, dy = \frac{y^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=2$$$:

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

Aplique a regra do múltiplo constante $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ usando $$$c=3$$$ e $$$f{\left(y \right)} = y^{5}$$$:

$$\frac{y^{3}}{3} - {\color{red}{\int{3 y^{5} d y}}} = \frac{y^{3}}{3} - {\color{red}{\left(3 \int{y^{5} d y}\right)}}$$

Aplique a regra da potência $$$\int y^{n}\, dy = \frac{y^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=5$$$:

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

Portanto,

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

Adicione a constante de integração:

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

$$$\int \left(- 3 y^{5} + y^{2}\right)\, dy = \left(- \frac{y^{6}}{2} + \frac{y^{3}}{3}\right) + C$$$A