Integralen av $$$\sin{\left(x \right)} + 18$$$

Bestäm $$$\int \left(\sin{\left(x \right)} + 18\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(\sin{\left(x \right)} + 18\right)d x}}} = {\color{red}{\left(\int{18 d x} + \int{\sin{\left(x \right)} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=18$$$:

$$\int{\sin{\left(x \right)} d x} + {\color{red}{\int{18 d x}}} = \int{\sin{\left(x \right)} d x} + {\color{red}{\left(18 x\right)}}$$

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

$$18 x + {\color{red}{\int{\sin{\left(x \right)} d x}}} = 18 x + {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

Alltså,

$$\int{\left(\sin{\left(x \right)} + 18\right)d x} = 18 x - \cos{\left(x \right)}$$

Lägg till integrationskonstanten:

$$\int{\left(\sin{\left(x \right)} + 18\right)d x} = 18 x - \cos{\left(x \right)}+C$$

$$$\int \left(\sin{\left(x \right)} + 18\right)\, dx = \left(18 x - \cos{\left(x \right)}\right) + C$$$A