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

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

Simplify the integrand:

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

For the integral $$$\int{v \cos{\left(v \right)} d v}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dm} = \operatorname{u}\operatorname{m} - \int \operatorname{m} \operatorname{du}$$$.

Let $$$\operatorname{u}=v$$$ and $$$\operatorname{dm}=\cos{\left(v \right)} dv$$$.

Then $$$\operatorname{du}=\left(v\right)^{\prime }dv=1 dv$$$ (steps can be seen ») and $$$\operatorname{m}=\int{\cos{\left(v \right)} d v}=\sin{\left(v \right)}$$$ (steps can be seen »).

Thus,

$${\color{red}{\int{v \cos{\left(v \right)} d v}}}={\color{red}{\left(v \cdot \sin{\left(v \right)}-\int{\sin{\left(v \right)} \cdot 1 d v}\right)}}={\color{red}{\left(v \sin{\left(v \right)} - \int{\sin{\left(v \right)} d v}\right)}}$$

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

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

Therefore,

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

Add the constant of integration:

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

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