Integral of $$$5 \sin{\left(x \right)}$$$

Find $$$\int 5 \sin{\left(x \right)}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=5$$$ and $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:

$${\color{red}{\int{5 \sin{\left(x \right)} d x}}} = {\color{red}{\left(5 \int{\sin{\left(x \right)} d x}\right)}}$$

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

$$5 {\color{red}{\int{\sin{\left(x \right)} d x}}} = 5 {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

Therefore,

$$\int{5 \sin{\left(x \right)} d x} = - 5 \cos{\left(x \right)}$$

Add the constant of integration:

$$\int{5 \sin{\left(x \right)} d x} = - 5 \cos{\left(x \right)}+C$$

$$$\int 5 \sin{\left(x \right)}\, dx = - 5 \cos{\left(x \right)} + C$$$A