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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=-1$$$ and $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:

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

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

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

Therefore,

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

Add the constant of integration:

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

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