Integralen av $$$\sin{\left(x \right)} - \pi$$$

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

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=\pi$$$:

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

Integralen av sinus är $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

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

Alltså,

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

Lägg till integrationskonstanten:

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

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