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

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

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(x \right)} = \sec^{2}{\left(\frac{x}{2} \right)}$$$:

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

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,

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

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

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

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

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

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

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

Portanto,

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

Adicione a constante de integração:

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

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