Integral of $$$\csc^{2}{\left(3 x \right)}$$$

Find $$$\int \csc^{2}{\left(3 x \right)}\, dx$$$.

Let $$$u=3 x$$$.

Then $$$du=\left(3 x\right)^{\prime }dx = 3 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{3}$$$.

Therefore,

$${\color{red}{\int{\csc^{2}{\left(3 x \right)} d x}}} = {\color{red}{\int{\frac{\csc^{2}{\left(u \right)}}{3} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{3}$$$ and $$$f{\left(u \right)} = \csc^{2}{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\csc^{2}{\left(u \right)}}{3} d u}}} = {\color{red}{\left(\frac{\int{\csc^{2}{\left(u \right)} d u}}{3}\right)}}$$

The integral of $$$\csc^{2}{\left(u \right)}$$$ is $$$\int{\csc^{2}{\left(u \right)} d u} = - \cot{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\csc^{2}{\left(u \right)} d u}}}}{3} = \frac{{\color{red}{\left(- \cot{\left(u \right)}\right)}}}{3}$$

Recall that $$$u=3 x$$$:

$$- \frac{\cot{\left({\color{red}{u}} \right)}}{3} = - \frac{\cot{\left({\color{red}{\left(3 x\right)}} \right)}}{3}$$

Therefore,

$$\int{\csc^{2}{\left(3 x \right)} d x} = - \frac{\cot{\left(3 x \right)}}{3}$$

Add the constant of integration:

$$\int{\csc^{2}{\left(3 x \right)} d x} = - \frac{\cot{\left(3 x \right)}}{3}+C$$

$$$\int \csc^{2}{\left(3 x \right)}\, dx = - \frac{\cot{\left(3 x \right)}}{3} + C$$$A