Integral de $$$\sec^{2}{\left(\frac{x}{2} \right)}$$$

Encontre $$$\int \sec^{2}{\left(\frac{x}{2} \right)}\, dx$$$.

Seja $$$u=\frac{x}{2}$$$.

Então $$$du=\left(\frac{x}{2}\right)^{\prime }dx = \frac{dx}{2}$$$ (veja os passos »), e obtemos $$$dx = 2 du$$$.

Portanto,

$${\color{red}{\int{\sec^{2}{\left(\frac{x}{2} \right)} d x}}} = {\color{red}{\int{2 \sec^{2}{\left(u \right)} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=2$$$ e $$$f{\left(u \right)} = \sec^{2}{\left(u \right)}$$$:

$${\color{red}{\int{2 \sec^{2}{\left(u \right)} d u}}} = {\color{red}{\left(2 \int{\sec^{2}{\left(u \right)} d u}\right)}}$$

A integral de $$$\sec^{2}{\left(u \right)}$$$ é $$$\int{\sec^{2}{\left(u \right)} d u} = \tan{\left(u \right)}$$$:

$$2 {\color{red}{\int{\sec^{2}{\left(u \right)} d u}}} = 2 {\color{red}{\tan{\left(u \right)}}}$$

Recorde que $$$u=\frac{x}{2}$$$:

$$2 \tan{\left({\color{red}{u}} \right)} = 2 \tan{\left({\color{red}{\left(\frac{x}{2}\right)}} \right)}$$

Portanto,

$$\int{\sec^{2}{\left(\frac{x}{2} \right)} d x} = 2 \tan{\left(\frac{x}{2} \right)}$$

Adicione a constante de integração:

$$\int{\sec^{2}{\left(\frac{x}{2} \right)} d x} = 2 \tan{\left(\frac{x}{2} \right)}+C$$

$$$\int \sec^{2}{\left(\frac{x}{2} \right)}\, dx = 2 \tan{\left(\frac{x}{2} \right)} + C$$$A