Integral von $$$x^{2} \left(2 x - 2\right) - 81 x$$$

Bestimme $$$\int \left(x^{2} \left(2 x - 2\right) - 81 x\right)\, dx$$$.

Gliedweise integrieren:

$${\color{red}{\int{\left(x^{2} \left(2 x - 2\right) - 81 x\right)d x}}} = {\color{red}{\left(- \int{81 x d x} + \int{x^{2} \left(2 x - 2\right) d x}\right)}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=81$$$ und $$$f{\left(x \right)} = x$$$ an:

$$\int{x^{2} \left(2 x - 2\right) d x} - {\color{red}{\int{81 x d x}}} = \int{x^{2} \left(2 x - 2\right) d x} - {\color{red}{\left(81 \int{x d x}\right)}}$$

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

$$\int{x^{2} \left(2 x - 2\right) d x} - 81 {\color{red}{\int{x d x}}}=\int{x^{2} \left(2 x - 2\right) d x} - 81 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\int{x^{2} \left(2 x - 2\right) d x} - 81 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Den Integranden vereinfachen:

$$- \frac{81 x^{2}}{2} + {\color{red}{\int{x^{2} \left(2 x - 2\right) d x}}} = - \frac{81 x^{2}}{2} + {\color{red}{\int{2 x^{2} \left(x - 1\right) d x}}}$$

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

$$- \frac{81 x^{2}}{2} + {\color{red}{\int{2 x^{2} \left(x - 1\right) d x}}} = - \frac{81 x^{2}}{2} + {\color{red}{\left(2 \int{x^{2} \left(x - 1\right) d x}\right)}}$$

Expand the expression:

$$- \frac{81 x^{2}}{2} + 2 {\color{red}{\int{x^{2} \left(x - 1\right) d x}}} = - \frac{81 x^{2}}{2} + 2 {\color{red}{\int{\left(x^{3} - x^{2}\right)d x}}}$$

Gliedweise integrieren:

$$- \frac{81 x^{2}}{2} + 2 {\color{red}{\int{\left(x^{3} - x^{2}\right)d x}}} = - \frac{81 x^{2}}{2} + 2 {\color{red}{\left(- \int{x^{2} d x} + \int{x^{3} d x}\right)}}$$

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=3$$$ an:

$$- \frac{81 x^{2}}{2} - 2 \int{x^{2} d x} + 2 {\color{red}{\int{x^{3} d x}}}=- \frac{81 x^{2}}{2} - 2 \int{x^{2} d x} + 2 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- \frac{81 x^{2}}{2} - 2 \int{x^{2} d x} + 2 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=2$$$ an:

$$\frac{x^{4}}{2} - \frac{81 x^{2}}{2} - 2 {\color{red}{\int{x^{2} d x}}}=\frac{x^{4}}{2} - \frac{81 x^{2}}{2} - 2 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{x^{4}}{2} - \frac{81 x^{2}}{2} - 2 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Daher,

$$\int{\left(x^{2} \left(2 x - 2\right) - 81 x\right)d x} = \frac{x^{4}}{2} - \frac{2 x^{3}}{3} - \frac{81 x^{2}}{2}$$

Vereinfachen:

$$\int{\left(x^{2} \left(2 x - 2\right) - 81 x\right)d x} = \frac{x^{2} \left(3 x^{2} - 4 x - 243\right)}{6}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(x^{2} \left(2 x - 2\right) - 81 x\right)d x} = \frac{x^{2} \left(3 x^{2} - 4 x - 243\right)}{6}+C$$

$$$\int \left(x^{2} \left(2 x - 2\right) - 81 x\right)\, dx = \frac{x^{2} \left(3 x^{2} - 4 x - 243\right)}{6} + C$$$A