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Integral de $$$- 2 x^{3} + x^{2} - 6 x$$$
Calculadora relacionada: Calculadora de Integrais Definidas e Impróprias
Sua entrada
Encontre $$$\int \left(- 2 x^{3} + x^{2} - 6 x\right)\, dx$$$.
Solução
Integre termo a termo:
$${\color{red}{\int{\left(- 2 x^{3} + x^{2} - 6 x\right)d x}}} = {\color{red}{\left(- \int{6 x d x} + \int{x^{2} d x} - \int{2 x^{3} d x}\right)}}$$
Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=2$$$:
$$- \int{6 x d x} - \int{2 x^{3} d x} + {\color{red}{\int{x^{2} d x}}}=- \int{6 x d x} - \int{2 x^{3} d x} + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \int{6 x d x} - \int{2 x^{3} d x} + {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$
Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=6$$$ e $$$f{\left(x \right)} = x$$$:
$$\frac{x^{3}}{3} - \int{2 x^{3} d x} - {\color{red}{\int{6 x d x}}} = \frac{x^{3}}{3} - \int{2 x^{3} d x} - {\color{red}{\left(6 \int{x d x}\right)}}$$
Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=1$$$:
$$\frac{x^{3}}{3} - \int{2 x^{3} d x} - 6 {\color{red}{\int{x d x}}}=\frac{x^{3}}{3} - \int{2 x^{3} d x} - 6 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\frac{x^{3}}{3} - \int{2 x^{3} d x} - 6 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$
Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=2$$$ e $$$f{\left(x \right)} = x^{3}$$$:
$$\frac{x^{3}}{3} - 3 x^{2} - {\color{red}{\int{2 x^{3} d x}}} = \frac{x^{3}}{3} - 3 x^{2} - {\color{red}{\left(2 \int{x^{3} d x}\right)}}$$
Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=3$$$:
$$\frac{x^{3}}{3} - 3 x^{2} - 2 {\color{red}{\int{x^{3} d x}}}=\frac{x^{3}}{3} - 3 x^{2} - 2 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=\frac{x^{3}}{3} - 3 x^{2} - 2 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$
Portanto,
$$\int{\left(- 2 x^{3} + x^{2} - 6 x\right)d x} = - \frac{x^{4}}{2} + \frac{x^{3}}{3} - 3 x^{2}$$
Simplifique:
$$\int{\left(- 2 x^{3} + x^{2} - 6 x\right)d x} = x^{2} \left(- \frac{x^{2}}{2} + \frac{x}{3} - 3\right)$$
Adicione a constante de integração:
$$\int{\left(- 2 x^{3} + x^{2} - 6 x\right)d x} = x^{2} \left(- \frac{x^{2}}{2} + \frac{x}{3} - 3\right)+C$$
Resposta
$$$\int \left(- 2 x^{3} + x^{2} - 6 x\right)\, dx = x^{2} \left(- \frac{x^{2}}{2} + \frac{x}{3} - 3\right) + C$$$A