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

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

Simplifique o integrando:

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

Para a integral $$$\int{v \cos{\left(v \right)} d v}$$$, use integração por partes $$$\int \operatorname{u} \operatorname{d\mu} = \operatorname{u}\operatorname{\mu} - \int \operatorname{\mu} \operatorname{du}$$$.

Sejam $$$\operatorname{u}=v$$$ e $$$\operatorname{d\mu}=\cos{\left(v \right)} dv$$$.

Então $$$\operatorname{du}=\left(v\right)^{\prime }dv=1 dv$$$ (os passos podem ser vistos ») e $$$\operatorname{\mu}=\int{\cos{\left(v \right)} d v}=\sin{\left(v \right)}$$$ (os passos podem ser vistos »).

Logo,

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

A integral do seno é $$$\int{\sin{\left(v \right)} d v} = - \cos{\left(v \right)}$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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