Integral von $$$\frac{\cos{\left(\frac{t}{2} \right)}}{2}$$$

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

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

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

Sei $$$u=\frac{t}{2}$$$.

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

Also,

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

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

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

Das Integral des Kosinus ist $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

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

Zur Erinnerung: $$$u=\frac{t}{2}$$$:

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{\cos{\left(\frac{t}{2} \right)}}{2} d t} = \sin{\left(\frac{t}{2} \right)}+C$$

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