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Integral de $$$- x^{2} + x$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \left(- x^{2} + x\right)\, dx$$$.
Solución
Integra término a término:
$${\color{red}{\int{\left(- x^{2} + x\right)d x}}} = {\color{red}{\left(\int{x d x} - \int{x^{2} d x}\right)}}$$
Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=1$$$:
$$- \int{x^{2} d x} + {\color{red}{\int{x d x}}}=- \int{x^{2} d x} + {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- \int{x^{2} d x} + {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$
Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=2$$$:
$$\frac{x^{2}}{2} - {\color{red}{\int{x^{2} d x}}}=\frac{x^{2}}{2} - {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{x^{2}}{2} - {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$
Por lo tanto,
$$\int{\left(- x^{2} + x\right)d x} = - \frac{x^{3}}{3} + \frac{x^{2}}{2}$$
Simplificar:
$$\int{\left(- x^{2} + x\right)d x} = \frac{x^{2} \left(3 - 2 x\right)}{6}$$
Añade la constante de integración:
$$\int{\left(- x^{2} + x\right)d x} = \frac{x^{2} \left(3 - 2 x\right)}{6}+C$$
Respuesta
$$$\int \left(- x^{2} + x\right)\, dx = \frac{x^{2} \left(3 - 2 x\right)}{6} + C$$$A