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Integralen av $$$- 4 x^{3} - 8 x^{2} - 4$$$
Relaterad kalkylator: Kalkylator för bestämda och oegentliga integraler
Din inmatning
Bestäm $$$\int \left(- 4 x^{3} - 8 x^{2} - 4\right)\, dx$$$.
Lösning
Integrera termvis:
$${\color{red}{\int{\left(- 4 x^{3} - 8 x^{2} - 4\right)d x}}} = {\color{red}{\left(- \int{4 d x} - \int{8 x^{2} d x} - \int{4 x^{3} d x}\right)}}$$
Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=4$$$:
$$- \int{8 x^{2} d x} - \int{4 x^{3} d x} - {\color{red}{\int{4 d x}}} = - \int{8 x^{2} d x} - \int{4 x^{3} d x} - {\color{red}{\left(4 x\right)}}$$
Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=8$$$ och $$$f{\left(x \right)} = x^{2}$$$:
$$- 4 x - \int{4 x^{3} d x} - {\color{red}{\int{8 x^{2} d x}}} = - 4 x - \int{4 x^{3} d x} - {\color{red}{\left(8 \int{x^{2} d x}\right)}}$$
Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=2$$$:
$$- 4 x - \int{4 x^{3} d x} - 8 {\color{red}{\int{x^{2} d x}}}=- 4 x - \int{4 x^{3} d x} - 8 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- 4 x - \int{4 x^{3} d x} - 8 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$
Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=4$$$ och $$$f{\left(x \right)} = x^{3}$$$:
$$- \frac{8 x^{3}}{3} - 4 x - {\color{red}{\int{4 x^{3} d x}}} = - \frac{8 x^{3}}{3} - 4 x - {\color{red}{\left(4 \int{x^{3} d x}\right)}}$$
Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=3$$$:
$$- \frac{8 x^{3}}{3} - 4 x - 4 {\color{red}{\int{x^{3} d x}}}=- \frac{8 x^{3}}{3} - 4 x - 4 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- \frac{8 x^{3}}{3} - 4 x - 4 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$
Alltså,
$$\int{\left(- 4 x^{3} - 8 x^{2} - 4\right)d x} = - x^{4} - \frac{8 x^{3}}{3} - 4 x$$
Förenkla:
$$\int{\left(- 4 x^{3} - 8 x^{2} - 4\right)d x} = - x \left(x^{3} + \frac{8 x^{2}}{3} + 4\right)$$
Lägg till integrationskonstanten:
$$\int{\left(- 4 x^{3} - 8 x^{2} - 4\right)d x} = - x \left(x^{3} + \frac{8 x^{2}}{3} + 4\right)+C$$
Svar
$$$\int \left(- 4 x^{3} - 8 x^{2} - 4\right)\, dx = - x \left(x^{3} + \frac{8 x^{2}}{3} + 4\right) + C$$$A