Integral de $$$3 \tan{\left(x \right)} \sec{\left(x \right)}$$$

Encontre $$$\int 3 \tan{\left(x \right)} \sec{\left(x \right)}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=3$$$ e $$$f{\left(x \right)} = \tan{\left(x \right)} \sec{\left(x \right)}$$$:

$${\color{red}{\int{3 \tan{\left(x \right)} \sec{\left(x \right)} d x}}} = {\color{red}{\left(3 \int{\tan{\left(x \right)} \sec{\left(x \right)} d x}\right)}}$$

A integral de $$$\tan{\left(x \right)} \sec{\left(x \right)}$$$ é $$$\int{\tan{\left(x \right)} \sec{\left(x \right)} d x} = \sec{\left(x \right)}$$$:

$$3 {\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}} = 3 {\color{red}{\sec{\left(x \right)}}}$$

Portanto,

$$\int{3 \tan{\left(x \right)} \sec{\left(x \right)} d x} = 3 \sec{\left(x \right)}$$

Adicione a constante de integração:

$$\int{3 \tan{\left(x \right)} \sec{\left(x \right)} d x} = 3 \sec{\left(x \right)}+C$$

$$$\int 3 \tan{\left(x \right)} \sec{\left(x \right)}\, dx = 3 \sec{\left(x \right)} + C$$$A