Integral von $$$\frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2}$$$

Bestimme $$$\int \frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2}\, dt$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ mit $$$c=\frac{1}{2}$$$ und $$$f{\left(t \right)} = \left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}$$$ an:

$${\color{red}{\int{\frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2} d t}}} = {\color{red}{\left(\frac{\int{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)} d t}}{2}\right)}}$$

Sei $$$u=1 - \frac{\sin{\left(t \right)}}{2}$$$.

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

Somit,

$$\frac{{\color{red}{\int{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)} d t}}}}{2} = \frac{{\color{red}{\int{\left(- 2 u^{2}\right)d u}}}}{2}$$

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

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

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

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

Zur Erinnerung: $$$u=1 - \frac{\sin{\left(t \right)}}{2}$$$:

$$- \frac{{\color{red}{u}}^{3}}{3} = - \frac{{\color{red}{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)}}^{3}}{3}$$

Daher,

$$\int{\frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2} d t} = - \frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{3}}{3}$$

Vereinfachen:

$$\int{\frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2} d t} = \frac{\left(\sin{\left(t \right)} - 2\right)^{3}}{24}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2} d t} = \frac{\left(\sin{\left(t \right)} - 2\right)^{3}}{24}+C$$

$$$\int \frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2}\, dt = \frac{\left(\sin{\left(t \right)} - 2\right)^{3}}{24} + C$$$A