Integral of $$$6 \cot{\left(x \right)} \csc{\left(x \right)}$$$

Find $$$\int 6 \cot{\left(x \right)} \csc{\left(x \right)}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=6$$$ and $$$f{\left(x \right)} = \cot{\left(x \right)} \csc{\left(x \right)}$$$:

$${\color{red}{\int{6 \cot{\left(x \right)} \csc{\left(x \right)} d x}}} = {\color{red}{\left(6 \int{\cot{\left(x \right)} \csc{\left(x \right)} d x}\right)}}$$

The integral of $$$\cot{\left(x \right)} \csc{\left(x \right)}$$$ is $$$\int{\cot{\left(x \right)} \csc{\left(x \right)} d x} = - \csc{\left(x \right)}$$$:

$$6 {\color{red}{\int{\cot{\left(x \right)} \csc{\left(x \right)} d x}}} = 6 {\color{red}{\left(- \csc{\left(x \right)}\right)}}$$

Therefore,

$$\int{6 \cot{\left(x \right)} \csc{\left(x \right)} d x} = - 6 \csc{\left(x \right)}$$

Add the constant of integration:

$$\int{6 \cot{\left(x \right)} \csc{\left(x \right)} d x} = - 6 \csc{\left(x \right)}+C$$

$$$\int 6 \cot{\left(x \right)} \csc{\left(x \right)}\, dx = - 6 \csc{\left(x \right)} + C$$$A