Integral of $$$4 \sin{\left(x \right)}$$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=4$$$ and $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:

$${\color{red}{\int{4 \sin{\left(x \right)} d x}}} = {\color{red}{\left(4 \int{\sin{\left(x \right)} d x}\right)}}$$

The integral of the sine is $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

$$4 {\color{red}{\int{\sin{\left(x \right)} d x}}} = 4 {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

Therefore,

$$\int{4 \sin{\left(x \right)} d x} = - 4 \cos{\left(x \right)}$$

Add the constant of integration:

$$\int{4 \sin{\left(x \right)} d x} = - 4 \cos{\left(x \right)}+C$$

$$$\int 4 \sin{\left(x \right)}\, dx = - 4 \cos{\left(x \right)} + C$$$A