Integral de $$$\frac{\tan{\left(x \right)}}{\cos{\left(x \right)}}$$$

Encontre $$$\int \frac{\tan{\left(x \right)}}{\cos{\left(x \right)}}\, dx$$$.

Reescreva o integrando:

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

Seja $$$u=\sec{\left(x \right)}$$$.

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

Assim,

$${\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}} = {\color{red}{\int{1 d u}}}$$

Aplique a regra da constante $$$\int c\, du = c u$$$ usando $$$c=1$$$:

$${\color{red}{\int{1 d u}}} = {\color{red}{u}}$$

Recorde que $$$u=\sec{\left(x \right)}$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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