Integral of $$$b \sin{\left(x \right)}$$$ with respect to $$$x$$$

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

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

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

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

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

Therefore,

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

Add the constant of integration:

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

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