Integralen av $$$\cos{\left(x \right)} \tan{\left(x \right)}$$$

Bestäm $$$\int \cos{\left(x \right)} \tan{\left(x \right)}\, dx$$$.

Förenkla integranden:

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

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,

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

Lägg till integrationskonstanten:

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

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