Integraal van $$$\frac{1}{4} - 5 \cos{\left(x \right)}$$$

Bepaal $$$\int \left(\frac{1}{4} - 5 \cos{\left(x \right)}\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(\frac{1}{4} - 5 \cos{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{\frac{1}{4} d x} - \int{5 \cos{\left(x \right)} d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=\frac{1}{4}$$$:

$$- \int{5 \cos{\left(x \right)} d x} + {\color{red}{\int{\frac{1}{4} d x}}} = - \int{5 \cos{\left(x \right)} d x} + {\color{red}{\left(\frac{x}{4}\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=5$$$ en $$$f{\left(x \right)} = \cos{\left(x \right)}$$$:

$$\frac{x}{4} - {\color{red}{\int{5 \cos{\left(x \right)} d x}}} = \frac{x}{4} - {\color{red}{\left(5 \int{\cos{\left(x \right)} d x}\right)}}$$

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

$$\frac{x}{4} - 5 {\color{red}{\int{\cos{\left(x \right)} d x}}} = \frac{x}{4} - 5 {\color{red}{\sin{\left(x \right)}}}$$

Dus,

$$\int{\left(\frac{1}{4} - 5 \cos{\left(x \right)}\right)d x} = \frac{x}{4} - 5 \sin{\left(x \right)}$$

Voeg de integratieconstante toe:

$$\int{\left(\frac{1}{4} - 5 \cos{\left(x \right)}\right)d x} = \frac{x}{4} - 5 \sin{\left(x \right)}+C$$

$$$\int \left(\frac{1}{4} - 5 \cos{\left(x \right)}\right)\, dx = \left(\frac{x}{4} - 5 \sin{\left(x \right)}\right) + C$$$A