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Integral de $$$- x^{5} \left(2 x - 6\right)$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \left(- x^{5} \left(2 x - 6\right)\right)\, dx$$$.
Solución
La entrada se reescribe: $$$\int{\left(- x^{5} \left(2 x - 6\right)\right)d x}=\int{x^{5} \left(6 - 2 x\right) d x}$$$.
Simplificar el integrando:
$${\color{red}{\int{x^{5} \left(6 - 2 x\right) d x}}} = {\color{red}{\int{2 x^{5} \left(3 - x\right) d x}}}$$
Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=2$$$ y $$$f{\left(x \right)} = x^{5} \left(3 - x\right)$$$:
$${\color{red}{\int{2 x^{5} \left(3 - x\right) d x}}} = {\color{red}{\left(2 \int{x^{5} \left(3 - x\right) d x}\right)}}$$
Expand the expression:
$$2 {\color{red}{\int{x^{5} \left(3 - x\right) d x}}} = 2 {\color{red}{\int{\left(- x^{6} + 3 x^{5}\right)d x}}}$$
Integra término a término:
$$2 {\color{red}{\int{\left(- x^{6} + 3 x^{5}\right)d x}}} = 2 {\color{red}{\left(\int{3 x^{5} d x} - \int{x^{6} d x}\right)}}$$
Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=6$$$:
$$2 \int{3 x^{5} d x} - 2 {\color{red}{\int{x^{6} d x}}}=2 \int{3 x^{5} d x} - 2 {\color{red}{\frac{x^{1 + 6}}{1 + 6}}}=2 \int{3 x^{5} d x} - 2 {\color{red}{\left(\frac{x^{7}}{7}\right)}}$$
Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=3$$$ y $$$f{\left(x \right)} = x^{5}$$$:
$$- \frac{2 x^{7}}{7} + 2 {\color{red}{\int{3 x^{5} d x}}} = - \frac{2 x^{7}}{7} + 2 {\color{red}{\left(3 \int{x^{5} d x}\right)}}$$
Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=5$$$:
$$- \frac{2 x^{7}}{7} + 6 {\color{red}{\int{x^{5} d x}}}=- \frac{2 x^{7}}{7} + 6 {\color{red}{\frac{x^{1 + 5}}{1 + 5}}}=- \frac{2 x^{7}}{7} + 6 {\color{red}{\left(\frac{x^{6}}{6}\right)}}$$
Por lo tanto,
$$\int{x^{5} \left(6 - 2 x\right) d x} = - \frac{2 x^{7}}{7} + x^{6}$$
Simplificar:
$$\int{x^{5} \left(6 - 2 x\right) d x} = \frac{x^{6} \left(7 - 2 x\right)}{7}$$
Añade la constante de integración:
$$\int{x^{5} \left(6 - 2 x\right) d x} = \frac{x^{6} \left(7 - 2 x\right)}{7}+C$$
Respuesta
$$$\int \left(- x^{5} \left(2 x - 6\right)\right)\, dx = \frac{x^{6} \left(7 - 2 x\right)}{7} + C$$$A