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

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

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=3$$$:

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

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

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

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

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

Alltså,

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

Förenkla:

$$\int{\left(2 x^{2} - 3\right)d x} = \frac{x \left(2 x^{2} - 9\right)}{3}$$

Lägg till integrationskonstanten:

$$\int{\left(2 x^{2} - 3\right)d x} = \frac{x \left(2 x^{2} - 9\right)}{3}+C$$

$$$\int \left(2 x^{2} - 3\right)\, dx = \frac{x \left(2 x^{2} - 9\right)}{3} + C$$$A