Integral von $$$\left(2 x - 1\right)^{4}$$$

Bestimme $$$\int \left(2 x - 1\right)^{4}\, dx$$$.

Sei $$$u=2 x - 1$$$.

Dann $$$du=\left(2 x - 1\right)^{\prime }dx = 2 dx$$$ (die Schritte sind » zu sehen), und es gilt $$$dx = \frac{du}{2}$$$.

Somit,

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

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{2}$$$ und $$$f{\left(u \right)} = u^{4}$$$ an:

$${\color{red}{\int{\frac{u^{4}}{2} d u}}} = {\color{red}{\left(\frac{\int{u^{4} d u}}{2}\right)}}$$

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

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

Zur Erinnerung: $$$u=2 x - 1$$$:

$$\frac{{\color{red}{u}}^{5}}{10} = \frac{{\color{red}{\left(2 x - 1\right)}}^{5}}{10}$$

Daher,

$$\int{\left(2 x - 1\right)^{4} d x} = \frac{\left(2 x - 1\right)^{5}}{10}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(2 x - 1\right)^{4} d x} = \frac{\left(2 x - 1\right)^{5}}{10}+C$$

$$$\int \left(2 x - 1\right)^{4}\, dx = \frac{\left(2 x - 1\right)^{5}}{10} + C$$$A