Integral de $$$\tan{\left(5 x \right)} \sec{\left(5 x \right)}$$$

Encontre $$$\int \tan{\left(5 x \right)} \sec{\left(5 x \right)}\, dx$$$.

Seja $$$u=5 x$$$.

Então $$$du=\left(5 x\right)^{\prime }dx = 5 dx$$$ (veja os passos »), e obtemos $$$dx = \frac{du}{5}$$$.

Assim,

$${\color{red}{\int{\tan{\left(5 x \right)} \sec{\left(5 x \right)} d x}}} = {\color{red}{\int{\frac{\tan{\left(u \right)} \sec{\left(u \right)}}{5} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{5}$$$ e $$$f{\left(u \right)} = \tan{\left(u \right)} \sec{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\tan{\left(u \right)} \sec{\left(u \right)}}{5} d u}}} = {\color{red}{\left(\frac{\int{\tan{\left(u \right)} \sec{\left(u \right)} d u}}{5}\right)}}$$

A integral de $$$\tan{\left(u \right)} \sec{\left(u \right)}$$$ é $$$\int{\tan{\left(u \right)} \sec{\left(u \right)} d u} = \sec{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\tan{\left(u \right)} \sec{\left(u \right)} d u}}}}{5} = \frac{{\color{red}{\sec{\left(u \right)}}}}{5}$$

Recorde que $$$u=5 x$$$:

$$\frac{\sec{\left({\color{red}{u}} \right)}}{5} = \frac{\sec{\left({\color{red}{\left(5 x\right)}} \right)}}{5}$$

Portanto,

$$\int{\tan{\left(5 x \right)} \sec{\left(5 x \right)} d x} = \frac{\sec{\left(5 x \right)}}{5}$$

Adicione a constante de integração:

$$\int{\tan{\left(5 x \right)} \sec{\left(5 x \right)} d x} = \frac{\sec{\left(5 x \right)}}{5}+C$$

$$$\int \tan{\left(5 x \right)} \sec{\left(5 x \right)}\, dx = \frac{\sec{\left(5 x \right)}}{5} + C$$$A