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

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(x \right)} = \sec^{2}{\left(x \right)}$$$:

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

The integral of $$$\sec^{2}{\left(x \right)}$$$ is $$$\int{\sec^{2}{\left(x \right)} d x} = \tan{\left(x \right)}$$$:

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

Therefore,

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

Add the constant of integration:

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

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