Integralen av $$$x^{3} - x^{2} - 2 x$$$

Bestäm $$$\int \left(x^{3} - x^{2} - 2 x\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(x^{3} - x^{2} - 2 x\right)d x}}} = {\color{red}{\left(- \int{2 x d x} - \int{x^{2} d x} + \int{x^{3} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=3$$$:

$$- \int{2 x d x} - \int{x^{2} d x} + {\color{red}{\int{x^{3} d x}}}=- \int{2 x d x} - \int{x^{2} d x} + {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- \int{2 x d x} - \int{x^{2} d x} + {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=2$$$:

$$\frac{x^{4}}{4} - \int{2 x d x} - {\color{red}{\int{x^{2} d x}}}=\frac{x^{4}}{4} - \int{2 x d x} - {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{x^{4}}{4} - \int{2 x d x} - {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=2$$$ och $$$f{\left(x \right)} = x$$$:

$$\frac{x^{4}}{4} - \frac{x^{3}}{3} - {\color{red}{\int{2 x d x}}} = \frac{x^{4}}{4} - \frac{x^{3}}{3} - {\color{red}{\left(2 \int{x d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=1$$$:

$$\frac{x^{4}}{4} - \frac{x^{3}}{3} - 2 {\color{red}{\int{x d x}}}=\frac{x^{4}}{4} - \frac{x^{3}}{3} - 2 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\frac{x^{4}}{4} - \frac{x^{3}}{3} - 2 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Alltså,

$$\int{\left(x^{3} - x^{2} - 2 x\right)d x} = \frac{x^{4}}{4} - \frac{x^{3}}{3} - x^{2}$$

Förenkla:

$$\int{\left(x^{3} - x^{2} - 2 x\right)d x} = x^{2} \left(\frac{x^{2}}{4} - \frac{x}{3} - 1\right)$$

Lägg till integrationskonstanten:

$$\int{\left(x^{3} - x^{2} - 2 x\right)d x} = x^{2} \left(\frac{x^{2}}{4} - \frac{x}{3} - 1\right)+C$$

$$$\int \left(x^{3} - x^{2} - 2 x\right)\, dx = x^{2} \left(\frac{x^{2}}{4} - \frac{x}{3} - 1\right) + C$$$A