Integral of $$$- \cos{\left(t \right)}$$$

Find $$$\int \left(- \cos{\left(t \right)}\right)\, dt$$$.

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=-1$$$ and $$$f{\left(t \right)} = \cos{\left(t \right)}$$$:

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

The integral of the cosine is $$$\int{\cos{\left(t \right)} d t} = \sin{\left(t \right)}$$$:

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

Therefore,

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

Add the constant of integration:

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

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