Integral von $$$- x^{2} + x$$$

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

Gliedweise integrieren:

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

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

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

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

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

Daher,

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

Vereinfachen:

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

Fügen Sie die Integrationskonstante hinzu:

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

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