Integral of $$$\frac{1}{\sec{\left(v \right)}}$$$

Find $$$\int \frac{1}{\sec{\left(v \right)}}\, dv$$$.

Rewrite the integrand in terms of the cosine:

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

The integral of the cosine is $$$\int{\cos{\left(v \right)} d v} = \sin{\left(v \right)}$$$:

$${\color{red}{\int{\cos{\left(v \right)} d v}}} = {\color{red}{\sin{\left(v \right)}}}$$

Therefore,

$$\int{\frac{1}{\sec{\left(v \right)}} d v} = \sin{\left(v \right)}$$

Add the constant of integration:

$$\int{\frac{1}{\sec{\left(v \right)}} d v} = \sin{\left(v \right)}+C$$

$$$\int \frac{1}{\sec{\left(v \right)}}\, dv = \sin{\left(v \right)} + C$$$A