Integral von $$$\frac{5 x^{2}}{2} - 3 x - \frac{21}{2}$$$

Bestimme $$$\int \left(\frac{5 x^{2}}{2} - 3 x - \frac{21}{2}\right)\, dx$$$.

Gliedweise integrieren:

$${\color{red}{\int{\left(\frac{5 x^{2}}{2} - 3 x - \frac{21}{2}\right)d x}}} = {\color{red}{\left(- \int{\frac{21}{2} d x} - \int{3 x d x} + \int{\frac{5 x^{2}}{2} d x}\right)}}$$

Wenden Sie die Konstantenregel $$$\int c\, dx = c x$$$ mit $$$c=\frac{21}{2}$$$ an:

$$- \int{3 x d x} + \int{\frac{5 x^{2}}{2} d x} - {\color{red}{\int{\frac{21}{2} d x}}} = - \int{3 x d x} + \int{\frac{5 x^{2}}{2} d x} - {\color{red}{\left(\frac{21 x}{2}\right)}}$$

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

$$- \frac{21 x}{2} + \int{\frac{5 x^{2}}{2} d x} - {\color{red}{\int{3 x d x}}} = - \frac{21 x}{2} + \int{\frac{5 x^{2}}{2} d x} - {\color{red}{\left(3 \int{x d x}\right)}}$$

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

$$- \frac{21 x}{2} + \int{\frac{5 x^{2}}{2} d x} - 3 {\color{red}{\int{x d x}}}=- \frac{21 x}{2} + \int{\frac{5 x^{2}}{2} d x} - 3 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- \frac{21 x}{2} + \int{\frac{5 x^{2}}{2} d x} - 3 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

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

$$- \frac{3 x^{2}}{2} - \frac{21 x}{2} + {\color{red}{\int{\frac{5 x^{2}}{2} d x}}} = - \frac{3 x^{2}}{2} - \frac{21 x}{2} + {\color{red}{\left(\frac{5 \int{x^{2} d x}}{2}\right)}}$$

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

$$- \frac{3 x^{2}}{2} - \frac{21 x}{2} + \frac{5 {\color{red}{\int{x^{2} d x}}}}{2}=- \frac{3 x^{2}}{2} - \frac{21 x}{2} + \frac{5 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}}{2}=- \frac{3 x^{2}}{2} - \frac{21 x}{2} + \frac{5 {\color{red}{\left(\frac{x^{3}}{3}\right)}}}{2}$$

Daher,

$$\int{\left(\frac{5 x^{2}}{2} - 3 x - \frac{21}{2}\right)d x} = \frac{5 x^{3}}{6} - \frac{3 x^{2}}{2} - \frac{21 x}{2}$$

Vereinfachen:

$$\int{\left(\frac{5 x^{2}}{2} - 3 x - \frac{21}{2}\right)d x} = \frac{x \left(5 x^{2} - 9 x - 63\right)}{6}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(\frac{5 x^{2}}{2} - 3 x - \frac{21}{2}\right)d x} = \frac{x \left(5 x^{2} - 9 x - 63\right)}{6}+C$$

$$$\int \left(\frac{5 x^{2}}{2} - 3 x - \frac{21}{2}\right)\, dx = \frac{x \left(5 x^{2} - 9 x - 63\right)}{6} + C$$$A