Integral de $$$1 - \sec^{2}{\left(x \right)}$$$

Encontre $$$\int \left(1 - \sec^{2}{\left(x \right)}\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(1 - \sec^{2}{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{1 d x} - \int{\sec^{2}{\left(x \right)} d x}\right)}}$$

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=1$$$:

$$- \int{\sec^{2}{\left(x \right)} d x} + {\color{red}{\int{1 d x}}} = - \int{\sec^{2}{\left(x \right)} d x} + {\color{red}{x}}$$

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

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

Portanto,

$$\int{\left(1 - \sec^{2}{\left(x \right)}\right)d x} = x - \tan{\left(x \right)}$$

Adicione a constante de integração:

$$\int{\left(1 - \sec^{2}{\left(x \right)}\right)d x} = x - \tan{\left(x \right)}+C$$

$$$\int \left(1 - \sec^{2}{\left(x \right)}\right)\, dx = \left(x - \tan{\left(x \right)}\right) + C$$$A