Integral von $$$- 4 x^{4} + 5 x^{2} - 5 x - 2$$$

Bestimme $$$\int \left(- 4 x^{4} + 5 x^{2} - 5 x - 2\right)\, dx$$$.

Gliedweise integrieren:

$${\color{red}{\int{\left(- 4 x^{4} + 5 x^{2} - 5 x - 2\right)d x}}} = {\color{red}{\left(- \int{2 d x} - \int{5 x d x} + \int{5 x^{2} d x} - \int{4 x^{4} d x}\right)}}$$

Wenden Sie die Konstantenregel $$$\int c\, dx = c x$$$ mit $$$c=2$$$ an:

$$- \int{5 x d x} + \int{5 x^{2} d x} - \int{4 x^{4} d x} - {\color{red}{\int{2 d x}}} = - \int{5 x d x} + \int{5 x^{2} d x} - \int{4 x^{4} d x} - {\color{red}{\left(2 x\right)}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=5$$$ und $$$f{\left(x \right)} = x$$$ an:

$$- 2 x + \int{5 x^{2} d x} - \int{4 x^{4} d x} - {\color{red}{\int{5 x d x}}} = - 2 x + \int{5 x^{2} d x} - \int{4 x^{4} d x} - {\color{red}{\left(5 \int{x d x}\right)}}$$

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

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

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=4$$$ und $$$f{\left(x \right)} = x^{4}$$$ an:

$$- \frac{5 x^{2}}{2} - 2 x + \int{5 x^{2} d x} - {\color{red}{\int{4 x^{4} d x}}} = - \frac{5 x^{2}}{2} - 2 x + \int{5 x^{2} d x} - {\color{red}{\left(4 \int{x^{4} d x}\right)}}$$

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=4$$$ an:

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

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=5$$$ und $$$f{\left(x \right)} = x^{2}$$$ an:

$$- \frac{4 x^{5}}{5} - \frac{5 x^{2}}{2} - 2 x + {\color{red}{\int{5 x^{2} d x}}} = - \frac{4 x^{5}}{5} - \frac{5 x^{2}}{2} - 2 x + {\color{red}{\left(5 \int{x^{2} d x}\right)}}$$

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

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

Daher,

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

Vereinfachen:

$$\int{\left(- 4 x^{4} + 5 x^{2} - 5 x - 2\right)d x} = \frac{x \left(- 24 x^{4} + 50 x^{2} - 75 x - 60\right)}{30}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(- 4 x^{4} + 5 x^{2} - 5 x - 2\right)d x} = \frac{x \left(- 24 x^{4} + 50 x^{2} - 75 x - 60\right)}{30}+C$$

$$$\int \left(- 4 x^{4} + 5 x^{2} - 5 x - 2\right)\, dx = \frac{x \left(- 24 x^{4} + 50 x^{2} - 75 x - 60\right)}{30} + C$$$A