Integraal van $$$\sin{\left(x \right)} - \pi$$$

Bepaal $$$\int \left(\sin{\left(x \right)} - \pi\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(\sin{\left(x \right)} - \pi\right)d x}}} = {\color{red}{\left(- \int{\pi d x} + \int{\sin{\left(x \right)} d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=\pi$$$:

$$\int{\sin{\left(x \right)} d x} - {\color{red}{\int{\pi d x}}} = \int{\sin{\left(x \right)} d x} - {\color{red}{\pi x}}$$

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

$$- \pi x + {\color{red}{\int{\sin{\left(x \right)} d x}}} = - \pi x + {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

Dus,

$$\int{\left(\sin{\left(x \right)} - \pi\right)d x} = - \pi x - \cos{\left(x \right)}$$

Voeg de integratieconstante toe:

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

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