Integral de $$$- \frac{3 x^{2}}{2} + 3 x$$$

Halla $$$\int \left(- \frac{3 x^{2}}{2} + 3 x\right)\, dx$$$.

Integra término a término:

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

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=3$$$ y $$$f{\left(x \right)} = x$$$:

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

Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=1$$$:

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

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=\frac{3}{2}$$$ y $$$f{\left(x \right)} = x^{2}$$$:

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

Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=2$$$:

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

Por lo tanto,

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

Simplificar:

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

Añade la constante de integración:

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

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