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

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=-1$$$ och $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:

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

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

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

Alltså,

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

Lägg till integrationskonstanten:

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

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