Integralen av $$$- x^{23} + x + 1$$$

Bestäm $$$\int \left(- x^{23} + x + 1\right)\, dx$$$.

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=1$$$:

$$\int{x d x} - \int{x^{23} d x} + {\color{red}{\int{1 d x}}} = \int{x d x} - \int{x^{23} d x} + {\color{red}{x}}$$

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

$$x - \int{x^{23} d x} + {\color{red}{\int{x d x}}}=x - \int{x^{23} d x} + {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=x - \int{x^{23} d x} + {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

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

$$\frac{x^{2}}{2} + x - {\color{red}{\int{x^{23} d x}}}=\frac{x^{2}}{2} + x - {\color{red}{\frac{x^{1 + 23}}{1 + 23}}}=\frac{x^{2}}{2} + x - {\color{red}{\left(\frac{x^{24}}{24}\right)}}$$

Alltså,

$$\int{\left(- x^{23} + x + 1\right)d x} = - \frac{x^{24}}{24} + \frac{x^{2}}{2} + x$$

Förenkla:

$$\int{\left(- x^{23} + x + 1\right)d x} = \frac{x \left(- x^{23} + 12 x + 24\right)}{24}$$

Lägg till integrationskonstanten:

$$\int{\left(- x^{23} + x + 1\right)d x} = \frac{x \left(- x^{23} + 12 x + 24\right)}{24}+C$$

$$$\int \left(- x^{23} + x + 1\right)\, dx = \frac{x \left(- x^{23} + 12 x + 24\right)}{24} + C$$$A