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

Find $$$\int \frac{1}{\cos^{2}{\left(\theta \right)}}\, d\theta$$$.

Rewrite the integrand in terms of the secant:

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

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

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

Therefore,

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

Add the constant of integration:

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

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