Integral of $$$9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)}$$$ with respect to $$$x$$$

Find $$$\int 9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=9 i n t$$$ and $$$f{\left(x \right)} = \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)}$$$:

$${\color{red}{\int{9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} d x}}} = {\color{red}{\left(9 i n t \int{\sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} d x}\right)}}$$

Rewrite the integrand:

$$9 i n t {\color{red}{\int{\sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} d x}}} = 9 i n t {\color{red}{\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x}}}$$

Let $$$u=\sin{\left(x \right)}$$$.

Then $$$du=\left(\sin{\left(x \right)}\right)^{\prime }dx = \cos{\left(x \right)} dx$$$ (steps can be seen »), and we have that $$$\cos{\left(x \right)} dx = du$$$.

Therefore,

$$9 i n t {\color{red}{\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x}}} = 9 i n t {\color{red}{\int{\frac{1}{u} d u}}}$$

The integral of $$$\frac{1}{u}$$$ is $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$9 i n t {\color{red}{\int{\frac{1}{u} d u}}} = 9 i n t {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

Recall that $$$u=\sin{\left(x \right)}$$$:

$$9 i n t \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = 9 i n t \ln{\left(\left|{{\color{red}{\sin{\left(x \right)}}}}\right| \right)}$$

Therefore,

$$\int{9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} d x} = 9 i n t \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)}$$

Add the constant of integration:

$$\int{9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} d x} = 9 i n t \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)}+C$$

$$$\int 9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)}\, dx = 9 i n t \ln\left(\left|{\sin{\left(x \right)}}\right|\right) + C$$$A