Integralen av $$$3 x^{23} - 7$$$

Bestäm $$$\int \left(3 x^{23} - 7\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(3 x^{23} - 7\right)d x}}} = {\color{red}{\left(- \int{7 d x} + \int{3 x^{23} d x}\right)}}$$

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

$$\int{3 x^{23} d x} - {\color{red}{\int{7 d x}}} = \int{3 x^{23} d x} - {\color{red}{\left(7 x\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=3$$$ och $$$f{\left(x \right)} = x^{23}$$$:

$$- 7 x + {\color{red}{\int{3 x^{23} d x}}} = - 7 x + {\color{red}{\left(3 \int{x^{23} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=23$$$:

$$- 7 x + 3 {\color{red}{\int{x^{23} d x}}}=- 7 x + 3 {\color{red}{\frac{x^{1 + 23}}{1 + 23}}}=- 7 x + 3 {\color{red}{\left(\frac{x^{24}}{24}\right)}}$$

Alltså,

$$\int{\left(3 x^{23} - 7\right)d x} = \frac{x^{24}}{8} - 7 x$$

Förenkla:

$$\int{\left(3 x^{23} - 7\right)d x} = \frac{x \left(x^{23} - 56\right)}{8}$$

Lägg till integrationskonstanten:

$$\int{\left(3 x^{23} - 7\right)d x} = \frac{x \left(x^{23} - 56\right)}{8}+C$$

$$$\int \left(3 x^{23} - 7\right)\, dx = \frac{x \left(x^{23} - 56\right)}{8} + C$$$A