Integral of $$$2 \tan{\left(x \right)} \sec{\left(x \right)}$$$

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = \tan{\left(x \right)} \sec{\left(x \right)}$$$:

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

The integral of $$$\tan{\left(x \right)} \sec{\left(x \right)}$$$ is $$$\int{\tan{\left(x \right)} \sec{\left(x \right)} d x} = \sec{\left(x \right)}$$$:

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

Therefore,

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

Add the constant of integration:

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

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