Integralen av $$$\cos^{2}{\left(\theta \right)}$$$

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

Använd potensreduceringsformeln $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ med $$$\alpha=\theta$$$:

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(\theta \right)} = \cos{\left(2 \theta \right)} + 1$$$:

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

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, d\theta = c \theta$$$ med $$$c=1$$$:

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

Låt $$$u=2 \theta$$$ vara.

Då $$$du=\left(2 \theta\right)^{\prime }d\theta = 2 d\theta$$$ (stegen kan ses »), och vi har att $$$d\theta = \frac{du}{2}$$$.

Integralen blir

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

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

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

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

Kom ihåg att $$$u=2 \theta$$$:

$$\frac{\theta}{2} + \frac{\sin{\left({\color{red}{u}} \right)}}{4} = \frac{\theta}{2} + \frac{\sin{\left({\color{red}{\left(2 \theta\right)}} \right)}}{4}$$

Alltså,

$$\int{\cos^{2}{\left(\theta \right)} d \theta} = \frac{\theta}{2} + \frac{\sin{\left(2 \theta \right)}}{4}$$

Lägg till integrationskonstanten:

$$\int{\cos^{2}{\left(\theta \right)} d \theta} = \frac{\theta}{2} + \frac{\sin{\left(2 \theta \right)}}{4}+C$$

$$$\int \cos^{2}{\left(\theta \right)}\, d\theta = \left(\frac{\theta}{2} + \frac{\sin{\left(2 \theta \right)}}{4}\right) + C$$$A