Integral de $$$- 14 x + \sin{\left(x \right)}$$$

Encontre $$$\int \left(- 14 x + \sin{\left(x \right)}\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(- 14 x + \sin{\left(x \right)}\right)d x}}} = {\color{red}{\left(- \int{14 x d x} + \int{\sin{\left(x \right)} d x}\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=14$$$ e $$$f{\left(x \right)} = x$$$:

$$\int{\sin{\left(x \right)} d x} - {\color{red}{\int{14 x d x}}} = \int{\sin{\left(x \right)} d x} - {\color{red}{\left(14 \int{x d x}\right)}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=1$$$:

$$\int{\sin{\left(x \right)} d x} - 14 {\color{red}{\int{x d x}}}=\int{\sin{\left(x \right)} d x} - 14 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\int{\sin{\left(x \right)} d x} - 14 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

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

$$- 7 x^{2} + {\color{red}{\int{\sin{\left(x \right)} d x}}} = - 7 x^{2} + {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

Portanto,

$$\int{\left(- 14 x + \sin{\left(x \right)}\right)d x} = - 7 x^{2} - \cos{\left(x \right)}$$

Adicione a constante de integração:

$$\int{\left(- 14 x + \sin{\left(x \right)}\right)d x} = - 7 x^{2} - \cos{\left(x \right)}+C$$

$$$\int \left(- 14 x + \sin{\left(x \right)}\right)\, dx = \left(- 7 x^{2} - \cos{\left(x \right)}\right) + C$$$A