|
|
|
|
|
|
|
|
|
|
|
|
Integral de $$$x^{3} - x^{2}$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \left(x^{3} - x^{2}\right)\, dx$$$.
Solución
Integra término a término:
$${\color{red}{\int{\left(x^{3} - x^{2}\right)d x}}} = {\color{red}{\left(- \int{x^{2} d x} + \int{x^{3} d x}\right)}}$$
Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=3$$$:
$$- \int{x^{2} d x} + {\color{red}{\int{x^{3} d x}}}=- \int{x^{2} d x} + {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- \int{x^{2} d x} + {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$
Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=2$$$:
$$\frac{x^{4}}{4} - {\color{red}{\int{x^{2} d x}}}=\frac{x^{4}}{4} - {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{x^{4}}{4} - {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$
Por lo tanto,
$$\int{\left(x^{3} - x^{2}\right)d x} = \frac{x^{4}}{4} - \frac{x^{3}}{3}$$
Simplificar:
$$\int{\left(x^{3} - x^{2}\right)d x} = \frac{x^{3} \left(3 x - 4\right)}{12}$$
Añade la constante de integración:
$$\int{\left(x^{3} - x^{2}\right)d x} = \frac{x^{3} \left(3 x - 4\right)}{12}+C$$
Respuesta
$$$\int \left(x^{3} - x^{2}\right)\, dx = \frac{x^{3} \left(3 x - 4\right)}{12} + C$$$A