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Integral de $$$\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1$$$
Calculadora relacionada: Calculadora de Integrais Definidas e Impróprias
Sua entrada
Encontre $$$\int \left(\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1\right)\, dx$$$.
Solução
Integre termo a termo:
$${\color{red}{\int{\left(\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1\right)d x}}} = {\color{red}{\left(- \int{1 d x} - \int{4 x^{3} d x} + \int{\sqrt{2} x^{\frac{5}{2}} d x}\right)}}$$
Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=1$$$:
$$- \int{4 x^{3} d x} + \int{\sqrt{2} x^{\frac{5}{2}} d x} - {\color{red}{\int{1 d x}}} = - \int{4 x^{3} d x} + \int{\sqrt{2} x^{\frac{5}{2}} d x} - {\color{red}{x}}$$
Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=4$$$ e $$$f{\left(x \right)} = x^{3}$$$:
$$- x + \int{\sqrt{2} x^{\frac{5}{2}} d x} - {\color{red}{\int{4 x^{3} d x}}} = - x + \int{\sqrt{2} x^{\frac{5}{2}} d x} - {\color{red}{\left(4 \int{x^{3} d x}\right)}}$$
Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=3$$$:
$$- x + \int{\sqrt{2} x^{\frac{5}{2}} d x} - 4 {\color{red}{\int{x^{3} d x}}}=- x + \int{\sqrt{2} x^{\frac{5}{2}} d x} - 4 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- x + \int{\sqrt{2} x^{\frac{5}{2}} d x} - 4 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$
Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\sqrt{2}$$$ e $$$f{\left(x \right)} = x^{\frac{5}{2}}$$$:
$$- x^{4} - x + {\color{red}{\int{\sqrt{2} x^{\frac{5}{2}} d x}}} = - x^{4} - x + {\color{red}{\sqrt{2} \int{x^{\frac{5}{2}} d x}}}$$
Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=\frac{5}{2}$$$:
$$- x^{4} - x + \sqrt{2} {\color{red}{\int{x^{\frac{5}{2}} d x}}}=- x^{4} - x + \sqrt{2} {\color{red}{\frac{x^{1 + \frac{5}{2}}}{1 + \frac{5}{2}}}}=- x^{4} - x + \sqrt{2} {\color{red}{\left(\frac{2 x^{\frac{7}{2}}}{7}\right)}}$$
Portanto,
$$\int{\left(\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1\right)d x} = \frac{2 \sqrt{2} x^{\frac{7}{2}}}{7} - x^{4} - x$$
Adicione a constante de integração:
$$\int{\left(\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1\right)d x} = \frac{2 \sqrt{2} x^{\frac{7}{2}}}{7} - x^{4} - x+C$$
Resposta
$$$\int \left(\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1\right)\, dx = \left(\frac{2 \sqrt{2} x^{\frac{7}{2}}}{7} - x^{4} - x\right) + C$$$A