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

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

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

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

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

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

Therefore,

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

Add the constant of integration:

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

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