Integraal van $$$36 \cos^{2}{\left(\theta \right)}$$$

Bepaal $$$\int 36 \cos^{2}{\left(\theta \right)}\, d\theta$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ toe met $$$c=36$$$ en $$$f{\left(\theta \right)} = \cos^{2}{\left(\theta \right)}$$$:

$${\color{red}{\int{36 \cos^{2}{\left(\theta \right)} d \theta}}} = {\color{red}{\left(36 \int{\cos^{2}{\left(\theta \right)} d \theta}\right)}}$$

Pas de machtsreductieformule $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ toe met $$$\alpha=\theta$$$:

$$36 {\color{red}{\int{\cos^{2}{\left(\theta \right)} d \theta}}} = 36 {\color{red}{\int{\left(\frac{\cos{\left(2 \theta \right)}}{2} + \frac{1}{2}\right)d \theta}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(\theta \right)} = \cos{\left(2 \theta \right)} + 1$$$:

$$36 {\color{red}{\int{\left(\frac{\cos{\left(2 \theta \right)}}{2} + \frac{1}{2}\right)d \theta}}} = 36 {\color{red}{\left(\frac{\int{\left(\cos{\left(2 \theta \right)} + 1\right)d \theta}}{2}\right)}}$$

Integreer termgewijs:

$$18 {\color{red}{\int{\left(\cos{\left(2 \theta \right)} + 1\right)d \theta}}} = 18 {\color{red}{\left(\int{1 d \theta} + \int{\cos{\left(2 \theta \right)} d \theta}\right)}}$$

Pas de constantenregel $$$\int c\, d\theta = c \theta$$$ toe met $$$c=1$$$:

$$18 \int{\cos{\left(2 \theta \right)} d \theta} + 18 {\color{red}{\int{1 d \theta}}} = 18 \int{\cos{\left(2 \theta \right)} d \theta} + 18 {\color{red}{\theta}}$$

Zij $$$u=2 \theta$$$.

Dan $$$du=\left(2 \theta\right)^{\prime }d\theta = 2 d\theta$$$ (de stappen zijn te zien »), en dan geldt dat $$$d\theta = \frac{du}{2}$$$.

Dus,

$$18 \theta + 18 {\color{red}{\int{\cos{\left(2 \theta \right)} d \theta}}} = 18 \theta + 18 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$18 \theta + 18 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}} = 18 \theta + 18 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}$$

De integraal van de cosinus is $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$18 \theta + 9 {\color{red}{\int{\cos{\left(u \right)} d u}}} = 18 \theta + 9 {\color{red}{\sin{\left(u \right)}}}$$

We herinneren eraan dat $$$u=2 \theta$$$:

$$18 \theta + 9 \sin{\left({\color{red}{u}} \right)} = 18 \theta + 9 \sin{\left({\color{red}{\left(2 \theta\right)}} \right)}$$

Dus,

$$\int{36 \cos^{2}{\left(\theta \right)} d \theta} = 18 \theta + 9 \sin{\left(2 \theta \right)}$$

Voeg de integratieconstante toe:

$$\int{36 \cos^{2}{\left(\theta \right)} d \theta} = 18 \theta + 9 \sin{\left(2 \theta \right)}+C$$

$$$\int 36 \cos^{2}{\left(\theta \right)}\, d\theta = \left(18 \theta + 9 \sin{\left(2 \theta \right)}\right) + C$$$A