Integral von $$$x^{4} - 19 x^{2} - 14 x + 32$$$

Bestimme $$$\int \left(x^{4} - 19 x^{2} - 14 x + 32\right)\, dx$$$.

Gliedweise integrieren:

$${\color{red}{\int{\left(x^{4} - 19 x^{2} - 14 x + 32\right)d x}}} = {\color{red}{\left(\int{32 d x} - \int{14 x d x} - \int{19 x^{2} d x} + \int{x^{4} d x}\right)}}$$

Wenden Sie die Konstantenregel $$$\int c\, dx = c x$$$ mit $$$c=32$$$ an:

$$- \int{14 x d x} - \int{19 x^{2} d x} + \int{x^{4} d x} + {\color{red}{\int{32 d x}}} = - \int{14 x d x} - \int{19 x^{2} d x} + \int{x^{4} d x} + {\color{red}{\left(32 x\right)}}$$

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

$$32 x - \int{14 x d x} - \int{19 x^{2} d x} + {\color{red}{\int{x^{4} d x}}}=32 x - \int{14 x d x} - \int{19 x^{2} d x} + {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=32 x - \int{14 x d x} - \int{19 x^{2} d x} + {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=19$$$ und $$$f{\left(x \right)} = x^{2}$$$ an:

$$\frac{x^{5}}{5} + 32 x - \int{14 x d x} - {\color{red}{\int{19 x^{2} d x}}} = \frac{x^{5}}{5} + 32 x - \int{14 x d x} - {\color{red}{\left(19 \int{x^{2} d x}\right)}}$$

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

$$\frac{x^{5}}{5} + 32 x - \int{14 x d x} - 19 {\color{red}{\int{x^{2} d x}}}=\frac{x^{5}}{5} + 32 x - \int{14 x d x} - 19 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{x^{5}}{5} + 32 x - \int{14 x d x} - 19 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=14$$$ und $$$f{\left(x \right)} = x$$$ an:

$$\frac{x^{5}}{5} - \frac{19 x^{3}}{3} + 32 x - {\color{red}{\int{14 x d x}}} = \frac{x^{5}}{5} - \frac{19 x^{3}}{3} + 32 x - {\color{red}{\left(14 \int{x d x}\right)}}$$

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

$$\frac{x^{5}}{5} - \frac{19 x^{3}}{3} + 32 x - 14 {\color{red}{\int{x d x}}}=\frac{x^{5}}{5} - \frac{19 x^{3}}{3} + 32 x - 14 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\frac{x^{5}}{5} - \frac{19 x^{3}}{3} + 32 x - 14 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Daher,

$$\int{\left(x^{4} - 19 x^{2} - 14 x + 32\right)d x} = \frac{x^{5}}{5} - \frac{19 x^{3}}{3} - 7 x^{2} + 32 x$$

Vereinfachen:

$$\int{\left(x^{4} - 19 x^{2} - 14 x + 32\right)d x} = \frac{x \left(3 x^{4} - 95 x^{2} - 105 x + 480\right)}{15}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(x^{4} - 19 x^{2} - 14 x + 32\right)d x} = \frac{x \left(3 x^{4} - 95 x^{2} - 105 x + 480\right)}{15}+C$$

$$$\int \left(x^{4} - 19 x^{2} - 14 x + 32\right)\, dx = \frac{x \left(3 x^{4} - 95 x^{2} - 105 x + 480\right)}{15} + C$$$A