Integralen av $$$56 - 3 x^{23}$$$

Bestäm $$$\int \left(56 - 3 x^{23}\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(56 - 3 x^{23}\right)d x}}} = {\color{red}{\left(\int{56 d x} - \int{3 x^{23} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=56$$$:

$$- \int{3 x^{23} d x} + {\color{red}{\int{56 d x}}} = - \int{3 x^{23} d x} + {\color{red}{\left(56 x\right)}}$$

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

$$56 x - {\color{red}{\int{3 x^{23} d x}}} = 56 x - {\color{red}{\left(3 \int{x^{23} d x}\right)}}$$

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

$$56 x - 3 {\color{red}{\int{x^{23} d x}}}=56 x - 3 {\color{red}{\frac{x^{1 + 23}}{1 + 23}}}=56 x - 3 {\color{red}{\left(\frac{x^{24}}{24}\right)}}$$

Alltså,

$$\int{\left(56 - 3 x^{23}\right)d x} = - \frac{x^{24}}{8} + 56 x$$

Förenkla:

$$\int{\left(56 - 3 x^{23}\right)d x} = \frac{x \left(448 - x^{23}\right)}{8}$$

Lägg till integrationskonstanten:

$$\int{\left(56 - 3 x^{23}\right)d x} = \frac{x \left(448 - x^{23}\right)}{8}+C$$

$$$\int \left(56 - 3 x^{23}\right)\, dx = \frac{x \left(448 - x^{23}\right)}{8} + C$$$A