Integral de $$$\sin{\left(x \right)} + 1$$$

Encontre $$$\int \left(\sin{\left(x \right)} + 1\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(\sin{\left(x \right)} + 1\right)d x}}} = {\color{red}{\left(\int{1 d x} + \int{\sin{\left(x \right)} d x}\right)}}$$

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

$$\int{\sin{\left(x \right)} d x} + {\color{red}{\int{1 d x}}} = \int{\sin{\left(x \right)} d x} + {\color{red}{x}}$$

A integral do seno é $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

$$x + {\color{red}{\int{\sin{\left(x \right)} d x}}} = x + {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

Portanto,

$$\int{\left(\sin{\left(x \right)} + 1\right)d x} = x - \cos{\left(x \right)}$$

Adicione a constante de integração:

$$\int{\left(\sin{\left(x \right)} + 1\right)d x} = x - \cos{\left(x \right)}+C$$

$$$\int \left(\sin{\left(x \right)} + 1\right)\, dx = \left(x - \cos{\left(x \right)}\right) + C$$$A