Integraal van $$$n \tan{\left(x \right)} \sec{\left(x \right)}$$$ met betrekking tot $$$x$$$

Bepaal $$$\int n \tan{\left(x \right)} \sec{\left(x \right)}\, dx$$$.

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

$${\color{red}{\int{n \tan{\left(x \right)} \sec{\left(x \right)} d x}}} = {\color{red}{n \int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}}$$

De integraal van $$$\tan{\left(x \right)} \sec{\left(x \right)}$$$ is $$$\int{\tan{\left(x \right)} \sec{\left(x \right)} d x} = \sec{\left(x \right)}$$$:

$$n {\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}} = n {\color{red}{\sec{\left(x \right)}}}$$

Dus,

$$\int{n \tan{\left(x \right)} \sec{\left(x \right)} d x} = n \sec{\left(x \right)}$$

Voeg de integratieconstante toe:

$$\int{n \tan{\left(x \right)} \sec{\left(x \right)} d x} = n \sec{\left(x \right)}+C$$

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