Integral von $$$2 x^{3} - 3 x^{2} - 5$$$

Bestimme $$$\int \left(2 x^{3} - 3 x^{2} - 5\right)\, dx$$$.

Gliedweise integrieren:

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

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

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

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=3$$$ und $$$f{\left(x \right)} = x^{2}$$$ an:

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

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

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

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=2$$$ und $$$f{\left(x \right)} = x^{3}$$$ an:

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

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

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

Daher,

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

Vereinfachen:

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

Fügen Sie die Integrationskonstante hinzu:

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

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