Integral von $$$- 10 x^{53} + \frac{x^{6}}{2}$$$

Bestimme $$$\int \left(- 10 x^{53} + \frac{x^{6}}{2}\right)\, dx$$$.

Gliedweise integrieren:

$${\color{red}{\int{\left(- 10 x^{53} + \frac{x^{6}}{2}\right)d x}}} = {\color{red}{\left(\int{\frac{x^{6}}{2} d x} - \int{10 x^{53} d x}\right)}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=\frac{1}{2}$$$ und $$$f{\left(x \right)} = x^{6}$$$ an:

$$- \int{10 x^{53} d x} + {\color{red}{\int{\frac{x^{6}}{2} d x}}} = - \int{10 x^{53} d x} + {\color{red}{\left(\frac{\int{x^{6} d x}}{2}\right)}}$$

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=6$$$ an:

$$- \int{10 x^{53} d x} + \frac{{\color{red}{\int{x^{6} d x}}}}{2}=- \int{10 x^{53} d x} + \frac{{\color{red}{\frac{x^{1 + 6}}{1 + 6}}}}{2}=- \int{10 x^{53} d x} + \frac{{\color{red}{\left(\frac{x^{7}}{7}\right)}}}{2}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=10$$$ und $$$f{\left(x \right)} = x^{53}$$$ an:

$$\frac{x^{7}}{14} - {\color{red}{\int{10 x^{53} d x}}} = \frac{x^{7}}{14} - {\color{red}{\left(10 \int{x^{53} d x}\right)}}$$

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=53$$$ an:

$$\frac{x^{7}}{14} - 10 {\color{red}{\int{x^{53} d x}}}=\frac{x^{7}}{14} - 10 {\color{red}{\frac{x^{1 + 53}}{1 + 53}}}=\frac{x^{7}}{14} - 10 {\color{red}{\left(\frac{x^{54}}{54}\right)}}$$

Daher,

$$\int{\left(- 10 x^{53} + \frac{x^{6}}{2}\right)d x} = - \frac{5 x^{54}}{27} + \frac{x^{7}}{14}$$

Vereinfachen:

$$\int{\left(- 10 x^{53} + \frac{x^{6}}{2}\right)d x} = \frac{x^{7} \left(27 - 70 x^{47}\right)}{378}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(- 10 x^{53} + \frac{x^{6}}{2}\right)d x} = \frac{x^{7} \left(27 - 70 x^{47}\right)}{378}+C$$

$$$\int \left(- 10 x^{53} + \frac{x^{6}}{2}\right)\, dx = \frac{x^{7} \left(27 - 70 x^{47}\right)}{378} + C$$$A