Intégrale de $$$\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)}$$$

Déterminez $$$\int \sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)}\, dx$$$.

Appliquez la règle du facteur constant $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ avec $$$c=\sqrt{2}$$$ et $$$f{\left(x \right)} = \tan{\left(x \right)} \sec{\left(x \right)}$$$ :

$${\color{red}{\int{\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)} d x}}} = {\color{red}{\sqrt{2} \int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}}$$

L’intégrale de $$$\tan{\left(x \right)} \sec{\left(x \right)}$$$ est $$$\int{\tan{\left(x \right)} \sec{\left(x \right)} d x} = \sec{\left(x \right)}$$$ :

$$\sqrt{2} {\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}} = \sqrt{2} {\color{red}{\sec{\left(x \right)}}}$$

Par conséquent,

$$\int{\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)} d x} = \sqrt{2} \sec{\left(x \right)}$$

Ajouter la constante d'intégration :

$$\int{\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)} d x} = \sqrt{2} \sec{\left(x \right)}+C$$

$$$\int \sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)}\, dx = \sqrt{2} \sec{\left(x \right)} + C$$$A