Integral von $$$- x^{2} - 8 x + \frac{1}{5}$$$

Bestimme $$$\int \left(- x^{2} - 8 x + \frac{1}{5}\right)\, dx$$$.

Gliedweise integrieren:

$${\color{red}{\int{\left(- x^{2} - 8 x + \frac{1}{5}\right)d x}}} = {\color{red}{\left(\int{\frac{1}{5} d x} - \int{8 x d x} - \int{x^{2} d x}\right)}}$$

Wenden Sie die Konstantenregel $$$\int c\, dx = c x$$$ mit $$$c=\frac{1}{5}$$$ an:

$$- \int{8 x d x} - \int{x^{2} d x} + {\color{red}{\int{\frac{1}{5} d x}}} = - \int{8 x d x} - \int{x^{2} d x} + {\color{red}{\left(\frac{x}{5}\right)}}$$

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

$$\frac{x}{5} - \int{8 x d x} - {\color{red}{\int{x^{2} d x}}}=\frac{x}{5} - \int{8 x d x} - {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{x}{5} - \int{8 x d x} - {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=8$$$ und $$$f{\left(x \right)} = x$$$ an:

$$- \frac{x^{3}}{3} + \frac{x}{5} - {\color{red}{\int{8 x d x}}} = - \frac{x^{3}}{3} + \frac{x}{5} - {\color{red}{\left(8 \int{x d x}\right)}}$$

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

$$- \frac{x^{3}}{3} + \frac{x}{5} - 8 {\color{red}{\int{x d x}}}=- \frac{x^{3}}{3} + \frac{x}{5} - 8 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- \frac{x^{3}}{3} + \frac{x}{5} - 8 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Daher,

$$\int{\left(- x^{2} - 8 x + \frac{1}{5}\right)d x} = - \frac{x^{3}}{3} - 4 x^{2} + \frac{x}{5}$$

Vereinfachen:

$$\int{\left(- x^{2} - 8 x + \frac{1}{5}\right)d x} = \frac{x \left(- 5 x^{2} - 60 x + 3\right)}{15}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(- x^{2} - 8 x + \frac{1}{5}\right)d x} = \frac{x \left(- 5 x^{2} - 60 x + 3\right)}{15}+C$$

$$$\int \left(- x^{2} - 8 x + \frac{1}{5}\right)\, dx = \frac{x \left(- 5 x^{2} - 60 x + 3\right)}{15} + C$$$A