Integral of csc^4xdx dx

1. Rewrite the integrand:

   $$\csc^{4}{\left(x \right)} 1 = \left(\cot^{2}{\left(x \right)} + 1\right) \csc^{2}{\left(x \right)}$$

2. There are multiple ways to do this integral.

   Method #1

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

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

      $$\int \left(- u^{2} - 1\right)\, du$$

      1. Integrate term-by-term:

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

            $$\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$$

            1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

               $$\int u^{2}\, du = \frac{u^{3}}{3}$$

            So, the result is: $- \frac{u^{3}}{3}$

         1. The integral of a constant is the constant times the variable of integration:

            $$\int \left(-1\right)\, du = - u$$

         The result is: $- \frac{u^{3}}{3} - u$

      Now substitute $u$ back in:

      $$- \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(\cot^{2}{\left(x \right)} + 1\right) \csc^{2}{\left(x \right)} = \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)} + \csc^{2}{\left(x \right)}$$

   2. Integrate term-by-term:

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

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

         $$\int u^{2}\, du$$

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

            $$\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$$

            1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

               $$\int u^{2}\, du = \frac{u^{3}}{3}$$

            So, the result is: $- \frac{u^{3}}{3}$

         Now substitute $u$ back in:

         $$- \frac{\cot^{3}{\left(x \right)}}{3}$$

      1. $$\int \csc^{2}{\left(x \right)}\, dx = - \cot{\left(x \right)}$$

      The result is: $- \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}$

   Method #3

   1. Rewrite the integrand:

      $$\left(\cot^{2}{\left(x \right)} + 1\right) \csc^{2}{\left(x \right)} = \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)} + \csc^{2}{\left(x \right)}$$

   2. Integrate term-by-term:

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

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

         $$\int u^{2}\, du$$

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

            $$\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$$

            1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

               $$\int u^{2}\, du = \frac{u^{3}}{3}$$

            So, the result is: $- \frac{u^{3}}{3}$

         Now substitute $u$ back in:

         $$- \frac{\cot^{3}{\left(x \right)}}{3}$$

      1. $$\int \csc^{2}{\left(x \right)}\, dx = - \cot{\left(x \right)}$$

      The result is: $- \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}$

3. Add the constant of integration:

   $$- \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$- \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}+ \mathrm{constant}$$