Integralen av $$$8 x^{6} - 5$$$

Bestäm $$$\int \left(8 x^{6} - 5\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(8 x^{6} - 5\right)d x}}} = {\color{red}{\left(- \int{5 d x} + \int{8 x^{6} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=5$$$:

$$\int{8 x^{6} d x} - {\color{red}{\int{5 d x}}} = \int{8 x^{6} d x} - {\color{red}{\left(5 x\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=8$$$ och $$$f{\left(x \right)} = x^{6}$$$:

$$- 5 x + {\color{red}{\int{8 x^{6} d x}}} = - 5 x + {\color{red}{\left(8 \int{x^{6} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=6$$$:

$$- 5 x + 8 {\color{red}{\int{x^{6} d x}}}=- 5 x + 8 {\color{red}{\frac{x^{1 + 6}}{1 + 6}}}=- 5 x + 8 {\color{red}{\left(\frac{x^{7}}{7}\right)}}$$

Alltså,

$$\int{\left(8 x^{6} - 5\right)d x} = \frac{8 x^{7}}{7} - 5 x$$

Förenkla:

$$\int{\left(8 x^{6} - 5\right)d x} = \frac{x \left(8 x^{6} - 35\right)}{7}$$

Lägg till integrationskonstanten:

$$\int{\left(8 x^{6} - 5\right)d x} = \frac{x \left(8 x^{6} - 35\right)}{7}+C$$

$$$\int \left(8 x^{6} - 5\right)\, dx = \frac{x \left(8 x^{6} - 35\right)}{7} + C$$$A