Integral of csc(x) dx

1. Rewrite the integrand:

   $$\csc{\left(x \right)} = \frac{\cot{\left(x \right)} \csc{\left(x \right)} + \csc^{2}{\left(x \right)}}{\cot{\left(x \right)} + \csc{\left(x \right)}}$$

2. Let $u = \cot{\left(x \right)} + \csc{\left(x \right)}$.

   Then let $du = \left(- \cot^{2}{\left(x \right)} - \cot{\left(x \right)} \csc{\left(x \right)} - 1\right) dx$ and substitute $- du$:

   $$\int \left(- \frac{1}{u}\right)\, du$$

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \frac{1}{u}\, du = - \int \frac{1}{u}\, du$$

      1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

      So, the result is: $- \log{\left(u \right)}$

   Now substitute $u$ back in:

   $$- \log{\left(\cot{\left(x \right)} + \csc{\left(x \right)} \right)}$$

3. Add the constant of integration:

   $$- \log{\left(\cot{\left(x \right)} + \csc{\left(x \right)} \right)}+ \mathrm{constant}$$

The answer is:

$$- \log{\left(\cot{\left(x \right)} + \csc{\left(x \right)} \right)}+ \mathrm{constant}$$