Integraal van $$$\frac{1}{2} - \frac{\sin{\left(2 a \right)}}{2}$$$

Bepaal $$$\int \left(\frac{1}{2} - \frac{\sin{\left(2 a \right)}}{2}\right)\, da$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(\frac{1}{2} - \frac{\sin{\left(2 a \right)}}{2}\right)d a}}} = {\color{red}{\left(\int{\frac{1}{2} d a} - \int{\frac{\sin{\left(2 a \right)}}{2} d a}\right)}}$$

Pas de constantenregel $$$\int c\, da = a c$$$ toe met $$$c=\frac{1}{2}$$$:

$$- \int{\frac{\sin{\left(2 a \right)}}{2} d a} + {\color{red}{\int{\frac{1}{2} d a}}} = - \int{\frac{\sin{\left(2 a \right)}}{2} d a} + {\color{red}{\left(\frac{a}{2}\right)}}$$

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

$$\frac{a}{2} - {\color{red}{\int{\frac{\sin{\left(2 a \right)}}{2} d a}}} = \frac{a}{2} - {\color{red}{\left(\frac{\int{\sin{\left(2 a \right)} d a}}{2}\right)}}$$

Zij $$$u=2 a$$$.

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

De integraal wordt

$$\frac{a}{2} - \frac{{\color{red}{\int{\sin{\left(2 a \right)} d a}}}}{2} = \frac{a}{2} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{2} d u}}}}{2}$$

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 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$$\frac{a}{2} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{2} d u}}}}{2} = \frac{a}{2} - \frac{{\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{2}\right)}}}{2}$$

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

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

We herinneren eraan dat $$$u=2 a$$$:

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

Dus,

$$\int{\left(\frac{1}{2} - \frac{\sin{\left(2 a \right)}}{2}\right)d a} = \frac{a}{2} + \frac{\cos{\left(2 a \right)}}{4}$$

Voeg de integratieconstante toe:

$$\int{\left(\frac{1}{2} - \frac{\sin{\left(2 a \right)}}{2}\right)d a} = \frac{a}{2} + \frac{\cos{\left(2 a \right)}}{4}+C$$

$$$\int \left(\frac{1}{2} - \frac{\sin{\left(2 a \right)}}{2}\right)\, da = \left(\frac{a}{2} + \frac{\cos{\left(2 a \right)}}{4}\right) + C$$$A