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

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

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

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

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

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

Therefore,

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

Add the constant of integration:

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

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