Integral von $$$\sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)}$$$

Bestimme $$$\int \sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)}\, dt$$$.

Schreiben Sie den Integranden mithilfe der Potenzreduktionsformeln $$$\sin^2\left( \alpha \right)=\frac{1}{2}-\frac{1}{2}\cos\left(2 \alpha \right)-$$$ mit $$$\alpha=t$$$ und $$$\cos^2\left( \beta \right)=\frac{1}{2}+\frac{1}{2}\cos\left(2 \beta \right)+$$$ mit $$$\beta=t$$$ um.:

$${\color{red}{\int{\sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)} d t}}} = {\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(2 t \right)}}{2}\right) \left(\frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}\right)^{2} d t}}}$$

Expandiere den Ausdruck:

$${\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(2 t \right)}}{2}\right) \left(\frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}\right)^{2} d t}}} = {\color{red}{\int{\left(- \frac{\cos^{3}{\left(2 t \right)}}{8} - \frac{\cos^{2}{\left(2 t \right)}}{8} + \frac{\cos{\left(2 t \right)}}{8} + \frac{1}{8}\right)d t}}}$$

Gliedweise integrieren:

$${\color{red}{\int{\left(- \frac{\cos^{3}{\left(2 t \right)}}{8} - \frac{\cos^{2}{\left(2 t \right)}}{8} + \frac{\cos{\left(2 t \right)}}{8} + \frac{1}{8}\right)d t}}} = {\color{red}{\left(\int{\frac{1}{8} d t} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{2}{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t}\right)}}$$

Wenden Sie die Konstantenregel $$$\int c\, dt = c t$$$ mit $$$c=\frac{1}{8}$$$ an:

$$\int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{2}{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} + {\color{red}{\int{\frac{1}{8} d t}}} = \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{2}{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} + {\color{red}{\left(\frac{t}{8}\right)}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ mit $$$c=\frac{1}{8}$$$ und $$$f{\left(t \right)} = \cos^{2}{\left(2 t \right)}$$$ an:

$$\frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - {\color{red}{\int{\frac{\cos^{2}{\left(2 t \right)}}{8} d t}}} = \frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - {\color{red}{\left(\frac{\int{\cos^{2}{\left(2 t \right)} d t}}{8}\right)}}$$

Sei $$$u=2 t$$$.

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

Also,

$$\frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\cos^{2}{\left(2 t \right)} d t}}}}{8} = \frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\frac{\cos^{2}{\left(u \right)}}{2} d u}}}}{8}$$

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

$$\frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\frac{\cos^{2}{\left(u \right)}}{2} d u}}}}{8} = \frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\left(\frac{\int{\cos^{2}{\left(u \right)} d u}}{2}\right)}}}{8}$$

Wende die Potenzreduktionsformel $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ mit $$$\alpha= u $$$ an:

$$\frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\cos^{2}{\left(u \right)} d u}}}}{16} = \frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{1}{2}\right)d u}}}}{16}$$

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

$$\frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{1}{2}\right)d u}}}}{16} = \frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\left(\frac{\int{\left(\cos{\left(2 u \right)} + 1\right)d u}}{2}\right)}}}{16}$$

Gliedweise integrieren:

$$\frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\left(\cos{\left(2 u \right)} + 1\right)d u}}}}{32} = \frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\left(\int{1 d u} + \int{\cos{\left(2 u \right)} d u}\right)}}}{32}$$

Wenden Sie die Konstantenregel $$$\int c\, du = c u$$$ mit $$$c=1$$$ an:

$$\frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{\int{\cos{\left(2 u \right)} d u}}{32} - \frac{{\color{red}{\int{1 d u}}}}{32} = \frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{\int{\cos{\left(2 u \right)} d u}}{32} - \frac{{\color{red}{u}}}{32}$$

Sei $$$v=2 u$$$.

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

Daher,

$$\frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\cos{\left(2 u \right)} d u}}}}{32} = \frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{32}$$

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

$$\frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{32} = \frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{2}\right)}}}{32}$$

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

$$\frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{64} = \frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\sin{\left(v \right)}}}}{64}$$

Zur Erinnerung: $$$v=2 u$$$:

$$\frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{\sin{\left({\color{red}{v}} \right)}}{64} = \frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{\sin{\left({\color{red}{\left(2 u\right)}} \right)}}{64}$$

Zur Erinnerung: $$$u=2 t$$$:

$$\frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{\sin{\left(2 {\color{red}{u}} \right)}}{64} - \frac{{\color{red}{u}}}{32} = \frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{\sin{\left(2 {\color{red}{\left(2 t\right)}} \right)}}{64} - \frac{{\color{red}{\left(2 t\right)}}}{32}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ mit $$$c=\frac{1}{8}$$$ und $$$f{\left(t \right)} = \cos^{3}{\left(2 t \right)}$$$ an:

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - {\color{red}{\int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t}}} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - {\color{red}{\left(\frac{\int{\cos^{3}{\left(2 t \right)} d t}}{8}\right)}}$$

Sei $$$u=2 t$$$.

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

Das Integral lässt sich umschreiben als

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\cos^{3}{\left(2 t \right)} d t}}}}{8} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\frac{\cos^{3}{\left(u \right)}}{2} d u}}}}{8}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{2}$$$ und $$$f{\left(u \right)} = \cos^{3}{\left(u \right)}$$$ an:

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\frac{\cos^{3}{\left(u \right)}}{2} d u}}}}{8} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\left(\frac{\int{\cos^{3}{\left(u \right)} d u}}{2}\right)}}}{8}$$

Klammern Sie einen Kosinus aus und drücken Sie alles Übrige in Abhängigkeit vom Sinus aus, mithilfe der Formel $$$\cos^2\left(\alpha \right)=-\sin^2\left(\alpha \right)+1$$$ mit $$$\alpha= u $$$.:

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\cos^{3}{\left(u \right)} d u}}}}{16} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\left(1 - \sin^{2}{\left(u \right)}\right) \cos{\left(u \right)} d u}}}}{16}$$

Sei $$$v=\sin{\left(u \right)}$$$.

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

Somit,

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\left(1 - \sin^{2}{\left(u \right)}\right) \cos{\left(u \right)} d u}}}}{16} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\left(1 - v^{2}\right)d v}}}}{16}$$

Gliedweise integrieren:

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\left(1 - v^{2}\right)d v}}}}{16} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\left(\int{1 d v} - \int{v^{2} d v}\right)}}}{16}$$

Wenden Sie die Konstantenregel $$$\int c\, dv = c v$$$ mit $$$c=1$$$ an:

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} + \frac{\int{v^{2} d v}}{16} - \frac{{\color{red}{\int{1 d v}}}}{16} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} + \frac{\int{v^{2} d v}}{16} - \frac{{\color{red}{v}}}{16}$$

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

$$\frac{t}{16} - \frac{v}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} + \frac{{\color{red}{\int{v^{2} d v}}}}{16}=\frac{t}{16} - \frac{v}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} + \frac{{\color{red}{\frac{v^{1 + 2}}{1 + 2}}}}{16}=\frac{t}{16} - \frac{v}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} + \frac{{\color{red}{\left(\frac{v^{3}}{3}\right)}}}{16}$$

Zur Erinnerung: $$$v=\sin{\left(u \right)}$$$:

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{v}}}{16} + \frac{{\color{red}{v}}^{3}}{48} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\sin{\left(u \right)}}}}{16} + \frac{{\color{red}{\sin{\left(u \right)}}}^{3}}{48}$$

Zur Erinnerung: $$$u=2 t$$$:

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{\sin{\left({\color{red}{u}} \right)}}{16} + \frac{\sin^{3}{\left({\color{red}{u}} \right)}}{48} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{\sin{\left({\color{red}{\left(2 t\right)}} \right)}}{16} + \frac{\sin^{3}{\left({\color{red}{\left(2 t\right)}} \right)}}{48}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ mit $$$c=\frac{1}{8}$$$ und $$$f{\left(t \right)} = \cos{\left(2 t \right)}$$$ an:

$$\frac{t}{16} + \frac{\sin^{3}{\left(2 t \right)}}{48} - \frac{\sin{\left(2 t \right)}}{16} - \frac{\sin{\left(4 t \right)}}{64} + {\color{red}{\int{\frac{\cos{\left(2 t \right)}}{8} d t}}} = \frac{t}{16} + \frac{\sin^{3}{\left(2 t \right)}}{48} - \frac{\sin{\left(2 t \right)}}{16} - \frac{\sin{\left(4 t \right)}}{64} + {\color{red}{\left(\frac{\int{\cos{\left(2 t \right)} d t}}{8}\right)}}$$

Das Integral $$$\int{\cos{\left(2 t \right)} d t}$$$ wurde bereits berechnet:

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

Daher,

$$\frac{t}{16} + \frac{\sin^{3}{\left(2 t \right)}}{48} - \frac{\sin{\left(2 t \right)}}{16} - \frac{\sin{\left(4 t \right)}}{64} + \frac{{\color{red}{\int{\cos{\left(2 t \right)} d t}}}}{8} = \frac{t}{16} + \frac{\sin^{3}{\left(2 t \right)}}{48} - \frac{\sin{\left(2 t \right)}}{16} - \frac{\sin{\left(4 t \right)}}{64} + \frac{{\color{red}{\left(\frac{\sin{\left(2 t \right)}}{2}\right)}}}{8}$$

Daher,

$$\int{\sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)} d t} = \frac{t}{16} + \frac{\sin^{3}{\left(2 t \right)}}{48} - \frac{\sin{\left(4 t \right)}}{64}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)} d t} = \frac{t}{16} + \frac{\sin^{3}{\left(2 t \right)}}{48} - \frac{\sin{\left(4 t \right)}}{64}+C$$

$$$\int \sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)}\, dt = \left(\frac{t}{16} + \frac{\sin^{3}{\left(2 t \right)}}{48} - \frac{\sin{\left(4 t \right)}}{64}\right) + C$$$A