Integraal van $$$\frac{1}{2} - \cos{\left(2 x \right)}$$$

Bepaal $$$\int \left(\frac{1}{2} - \cos{\left(2 x \right)}\right)\, dx$$$.

Integreer termgewijs:

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

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

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

Zij $$$u=2 x$$$.

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

Dus,

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

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

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

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

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

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

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

Dus,

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

Vereenvoudig:

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

Voeg de integratieconstante toe:

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

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