Integraal van $$$16 \cos{\left(x \right)} \tan{\left(x \right)}$$$

Bepaal $$$\int 16 \cos{\left(x \right)} \tan{\left(x \right)}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=16$$$ en $$$f{\left(x \right)} = \cos{\left(x \right)} \tan{\left(x \right)}$$$:

$${\color{red}{\int{16 \cos{\left(x \right)} \tan{\left(x \right)} d x}}} = {\color{red}{\left(16 \int{\cos{\left(x \right)} \tan{\left(x \right)} d x}\right)}}$$

Vereenvoudig de integraand:

$$16 {\color{red}{\int{\cos{\left(x \right)} \tan{\left(x \right)} d x}}} = 16 {\color{red}{\int{\sin{\left(x \right)} d x}}}$$

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

$$16 {\color{red}{\int{\sin{\left(x \right)} d x}}} = 16 {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

Dus,

$$\int{16 \cos{\left(x \right)} \tan{\left(x \right)} d x} = - 16 \cos{\left(x \right)}$$

Voeg de integratieconstante toe:

$$\int{16 \cos{\left(x \right)} \tan{\left(x \right)} d x} = - 16 \cos{\left(x \right)}+C$$

$$$\int 16 \cos{\left(x \right)} \tan{\left(x \right)}\, dx = - 16 \cos{\left(x \right)} + C$$$A