Integralen av $$$\cos{\left(6 \theta \right)}$$$

Bestäm $$$\int \cos{\left(6 \theta \right)}\, d\theta$$$.

Låt $$$u=6 \theta$$$ vara.

Då $$$du=\left(6 \theta\right)^{\prime }d\theta = 6 d\theta$$$ (stegen kan ses »), och vi har att $$$d\theta = \frac{du}{6}$$$.

Integralen blir

$${\color{red}{\int{\cos{\left(6 \theta \right)} d \theta}}} = {\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{6}$$$ och $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}} = {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{6}\right)}}$$

Integralen av cosinus är $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{6} = \frac{{\color{red}{\sin{\left(u \right)}}}}{6}$$

Kom ihåg att $$$u=6 \theta$$$:

$$\frac{\sin{\left({\color{red}{u}} \right)}}{6} = \frac{\sin{\left({\color{red}{\left(6 \theta\right)}} \right)}}{6}$$

Alltså,

$$\int{\cos{\left(6 \theta \right)} d \theta} = \frac{\sin{\left(6 \theta \right)}}{6}$$

Lägg till integrationskonstanten:

$$\int{\cos{\left(6 \theta \right)} d \theta} = \frac{\sin{\left(6 \theta \right)}}{6}+C$$

$$$\int \cos{\left(6 \theta \right)}\, d\theta = \frac{\sin{\left(6 \theta \right)}}{6} + C$$$A