Integral von $$$63 x^{23} - 126$$$

Bestimme $$$\int \left(63 x^{23} - 126\right)\, dx$$$.

Gliedweise integrieren:

$${\color{red}{\int{\left(63 x^{23} - 126\right)d x}}} = {\color{red}{\left(- \int{126 d x} + \int{63 x^{23} d x}\right)}}$$

Wenden Sie die Konstantenregel $$$\int c\, dx = c x$$$ mit $$$c=126$$$ an:

$$\int{63 x^{23} d x} - {\color{red}{\int{126 d x}}} = \int{63 x^{23} d x} - {\color{red}{\left(126 x\right)}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=63$$$ und $$$f{\left(x \right)} = x^{23}$$$ an:

$$- 126 x + {\color{red}{\int{63 x^{23} d x}}} = - 126 x + {\color{red}{\left(63 \int{x^{23} d x}\right)}}$$

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=23$$$ an:

$$- 126 x + 63 {\color{red}{\int{x^{23} d x}}}=- 126 x + 63 {\color{red}{\frac{x^{1 + 23}}{1 + 23}}}=- 126 x + 63 {\color{red}{\left(\frac{x^{24}}{24}\right)}}$$

Daher,

$$\int{\left(63 x^{23} - 126\right)d x} = \frac{21 x^{24}}{8} - 126 x$$

Vereinfachen:

$$\int{\left(63 x^{23} - 126\right)d x} = \frac{21 x \left(x^{23} - 48\right)}{8}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(63 x^{23} - 126\right)d x} = \frac{21 x \left(x^{23} - 48\right)}{8}+C$$

$$$\int \left(63 x^{23} - 126\right)\, dx = \frac{21 x \left(x^{23} - 48\right)}{8} + C$$$A