Integral von $$$- 3 x^{218} + x^{34} - 9$$$

Bestimme $$$\int \left(- 3 x^{218} + x^{34} - 9\right)\, dx$$$.

Gliedweise integrieren:

$${\color{red}{\int{\left(- 3 x^{218} + x^{34} - 9\right)d x}}} = {\color{red}{\left(- \int{9 d x} + \int{x^{34} d x} - \int{3 x^{218} d x}\right)}}$$

Wenden Sie die Konstantenregel $$$\int c\, dx = c x$$$ mit $$$c=9$$$ an:

$$\int{x^{34} d x} - \int{3 x^{218} d x} - {\color{red}{\int{9 d x}}} = \int{x^{34} d x} - \int{3 x^{218} d x} - {\color{red}{\left(9 x\right)}}$$

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=34$$$ an:

$$- 9 x - \int{3 x^{218} d x} + {\color{red}{\int{x^{34} d x}}}=- 9 x - \int{3 x^{218} d x} + {\color{red}{\frac{x^{1 + 34}}{1 + 34}}}=- 9 x - \int{3 x^{218} d x} + {\color{red}{\left(\frac{x^{35}}{35}\right)}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=3$$$ und $$$f{\left(x \right)} = x^{218}$$$ an:

$$\frac{x^{35}}{35} - 9 x - {\color{red}{\int{3 x^{218} d x}}} = \frac{x^{35}}{35} - 9 x - {\color{red}{\left(3 \int{x^{218} d x}\right)}}$$

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=218$$$ an:

$$\frac{x^{35}}{35} - 9 x - 3 {\color{red}{\int{x^{218} d x}}}=\frac{x^{35}}{35} - 9 x - 3 {\color{red}{\frac{x^{1 + 218}}{1 + 218}}}=\frac{x^{35}}{35} - 9 x - 3 {\color{red}{\left(\frac{x^{219}}{219}\right)}}$$

Daher,

$$\int{\left(- 3 x^{218} + x^{34} - 9\right)d x} = - \frac{x^{219}}{73} + \frac{x^{35}}{35} - 9 x$$

Vereinfachen:

$$\int{\left(- 3 x^{218} + x^{34} - 9\right)d x} = x \left(- \frac{x^{218}}{73} + \frac{x^{34}}{35} - 9\right)$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(- 3 x^{218} + x^{34} - 9\right)d x} = x \left(- \frac{x^{218}}{73} + \frac{x^{34}}{35} - 9\right)+C$$

$$$\int \left(- 3 x^{218} + x^{34} - 9\right)\, dx = x \left(- \frac{x^{218}}{73} + \frac{x^{34}}{35} - 9\right) + C$$$A