Integral von $$$\sin{\left(x \right)} \cos^{4}{\left(x \right)}$$$

Bestimme $$$\int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx$$$.

Sei $$$u=\cos{\left(x \right)}$$$.

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

Also,

$${\color{red}{\int{\sin{\left(x \right)} \cos^{4}{\left(x \right)} d x}}} = {\color{red}{\int{\left(- u^{4}\right)d u}}}$$

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

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

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

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

Zur Erinnerung: $$$u=\cos{\left(x \right)}$$$:

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

$$\int{\sin{\left(x \right)} \cos^{4}{\left(x \right)} d x} = - \frac{\cos^{5}{\left(x \right)}}{5}+C$$

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