Integralen av $$$x^{2} - 6 x - 7$$$

Bestäm $$$\int \left(x^{2} - 6 x - 7\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(x^{2} - 6 x - 7\right)d x}}} = {\color{red}{\left(- \int{7 d x} - \int{6 x d x} + \int{x^{2} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=7$$$:

$$- \int{6 x d x} + \int{x^{2} d x} - {\color{red}{\int{7 d x}}} = - \int{6 x d x} + \int{x^{2} d x} - {\color{red}{\left(7 x\right)}}$$

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

$$- 7 x - \int{6 x d x} + {\color{red}{\int{x^{2} d x}}}=- 7 x - \int{6 x d x} + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- 7 x - \int{6 x d x} + {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=6$$$ och $$$f{\left(x \right)} = x$$$:

$$\frac{x^{3}}{3} - 7 x - {\color{red}{\int{6 x d x}}} = \frac{x^{3}}{3} - 7 x - {\color{red}{\left(6 \int{x d x}\right)}}$$

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

$$\frac{x^{3}}{3} - 7 x - 6 {\color{red}{\int{x d x}}}=\frac{x^{3}}{3} - 7 x - 6 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\frac{x^{3}}{3} - 7 x - 6 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Alltså,

$$\int{\left(x^{2} - 6 x - 7\right)d x} = \frac{x^{3}}{3} - 3 x^{2} - 7 x$$

Förenkla:

$$\int{\left(x^{2} - 6 x - 7\right)d x} = \frac{x \left(x^{2} - 9 x - 21\right)}{3}$$

Lägg till integrationskonstanten:

$$\int{\left(x^{2} - 6 x - 7\right)d x} = \frac{x \left(x^{2} - 9 x - 21\right)}{3}+C$$

$$$\int \left(x^{2} - 6 x - 7\right)\, dx = \frac{x \left(x^{2} - 9 x - 21\right)}{3} + C$$$A