Integral von $$$\cos^{4}{\left(7 x \right)}$$$

Bestimme $$$\int \cos^{4}{\left(7 x \right)}\, dx$$$.

Sei $$$u=7 x$$$.

Dann $$$du=\left(7 x\right)^{\prime }dx = 7 dx$$$ (die Schritte sind » zu sehen), und es gilt $$$dx = \frac{du}{7}$$$.

Das Integral lässt sich umschreiben als

$${\color{red}{\int{\cos^{4}{\left(7 x \right)} d x}}} = {\color{red}{\int{\frac{\cos^{4}{\left(u \right)}}{7} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{7}$$$ und $$$f{\left(u \right)} = \cos^{4}{\left(u \right)}$$$ an:

$${\color{red}{\int{\frac{\cos^{4}{\left(u \right)}}{7} d u}}} = {\color{red}{\left(\frac{\int{\cos^{4}{\left(u \right)} d u}}{7}\right)}}$$

Wende die Potenzreduktionsformel $$$\cos^{4}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{\cos{\left(4 \alpha \right)}}{8} + \frac{3}{8}$$$ mit $$$\alpha= u $$$ an:

$$\frac{{\color{red}{\int{\cos^{4}{\left(u \right)} d u}}}}{7} = \frac{{\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{\cos{\left(4 u \right)}}{8} + \frac{3}{8}\right)d u}}}}{7}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{8}$$$ und $$$f{\left(u \right)} = 4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3$$$ an:

$$\frac{{\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{\cos{\left(4 u \right)}}{8} + \frac{3}{8}\right)d u}}}}{7} = \frac{{\color{red}{\left(\frac{\int{\left(4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3\right)d u}}{8}\right)}}}{7}$$

Gliedweise integrieren:

$$\frac{{\color{red}{\int{\left(4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3\right)d u}}}}{56} = \frac{{\color{red}{\left(\int{3 d u} + \int{4 \cos{\left(2 u \right)} d u} + \int{\cos{\left(4 u \right)} d u}\right)}}}{56}$$

Wenden Sie die Konstantenregel $$$\int c\, du = c u$$$ mit $$$c=3$$$ an:

$$\frac{\int{4 \cos{\left(2 u \right)} d u}}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{3 d u}}}}{56} = \frac{\int{4 \cos{\left(2 u \right)} d u}}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\left(3 u\right)}}}{56}$$

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

$$\frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{4 \cos{\left(2 u \right)} d u}}}}{56} = \frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\left(4 \int{\cos{\left(2 u \right)} d u}\right)}}}{56}$$

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

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

Somit,

$$\frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{\cos{\left(2 u \right)} d u}}}}{14} = \frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{14}$$

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:

$$\frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{14} = \frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{2}\right)}}}{14}$$

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

$$\frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{28} = \frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\sin{\left(v \right)}}}}{28}$$

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

$$\frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{\sin{\left({\color{red}{v}} \right)}}{28} = \frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{\sin{\left({\color{red}{\left(2 u\right)}} \right)}}{28}$$

Sei $$$v=4 u$$$.

Dann $$$dv=\left(4 u\right)^{\prime }du = 4 du$$$ (die Schritte sind » zu sehen), und es gilt $$$du = \frac{dv}{4}$$$.

Somit,

$$\frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\int{\cos{\left(4 u \right)} d u}}}}{56} = \frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}}}{56}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ mit $$$c=\frac{1}{4}$$$ und $$$f{\left(v \right)} = \cos{\left(v \right)}$$$ an:

$$\frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}}}{56} = \frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{4}\right)}}}{56}$$

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

$$\frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{224} = \frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\sin{\left(v \right)}}}}{224}$$

Zur Erinnerung: $$$v=4 u$$$:

$$\frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{\sin{\left({\color{red}{v}} \right)}}{224} = \frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{\sin{\left({\color{red}{\left(4 u\right)}} \right)}}{224}$$

Zur Erinnerung: $$$u=7 x$$$:

$$\frac{\sin{\left(2 {\color{red}{u}} \right)}}{28} + \frac{\sin{\left(4 {\color{red}{u}} \right)}}{224} + \frac{3 {\color{red}{u}}}{56} = \frac{\sin{\left(2 {\color{red}{\left(7 x\right)}} \right)}}{28} + \frac{\sin{\left(4 {\color{red}{\left(7 x\right)}} \right)}}{224} + \frac{3 {\color{red}{\left(7 x\right)}}}{56}$$

Daher,

$$\int{\cos^{4}{\left(7 x \right)} d x} = \frac{3 x}{8} + \frac{\sin{\left(14 x \right)}}{28} + \frac{\sin{\left(28 x \right)}}{224}$$

Vereinfachen:

$$\int{\cos^{4}{\left(7 x \right)} d x} = \frac{84 x + 8 \sin{\left(14 x \right)} + \sin{\left(28 x \right)}}{224}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\cos^{4}{\left(7 x \right)} d x} = \frac{84 x + 8 \sin{\left(14 x \right)} + \sin{\left(28 x \right)}}{224}+C$$

$$$\int \cos^{4}{\left(7 x \right)}\, dx = \frac{84 x + 8 \sin{\left(14 x \right)} + \sin{\left(28 x \right)}}{224} + C$$$A