Integral of $$$\frac{1}{\cos^{2}{\left(x \right)}}$$$

Rewrite the integrand in terms of the secant:

$${\color{red}{\int{\frac{1}{\cos^{2}{\left(x \right)}} d x}}} = {\color{red}{\int{\sec^{2}{\left(x \right)} d x}}}$$

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,

$$\int{\frac{1}{\cos^{2}{\left(x \right)}} d x} = \tan{\left(x \right)}$$

Add the constant of integration:

$$\int{\frac{1}{\cos^{2}{\left(x \right)}} d x} = \tan{\left(x \right)}+C$$

Answer: $$$\int{\frac{1}{\cos^{2}{\left(x \right)}} d x}=\tan{\left(x \right)}+C$$$