Integralen av $$$\frac{x^{2}}{2} - 2 x$$$

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

Integrera termvis:

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(x \right)} = x^{2}$$$:

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

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

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

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

$$\frac{x^{3}}{6} - {\color{red}{\int{2 x d x}}} = \frac{x^{3}}{6} - {\color{red}{\left(2 \int{x d x}\right)}}$$

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

$$\frac{x^{3}}{6} - 2 {\color{red}{\int{x d x}}}=\frac{x^{3}}{6} - 2 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\frac{x^{3}}{6} - 2 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Alltså,

$$\int{\left(\frac{x^{2}}{2} - 2 x\right)d x} = \frac{x^{3}}{6} - x^{2}$$

Förenkla:

$$\int{\left(\frac{x^{2}}{2} - 2 x\right)d x} = \frac{x^{2} \left(x - 6\right)}{6}$$

Lägg till integrationskonstanten:

$$\int{\left(\frac{x^{2}}{2} - 2 x\right)d x} = \frac{x^{2} \left(x - 6\right)}{6}+C$$

$$$\int \left(\frac{x^{2}}{2} - 2 x\right)\, dx = \frac{x^{2} \left(x - 6\right)}{6} + C$$$A