Integral of xcsc²x dx

1. Use integration by parts:

   $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

   Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \csc^{2}{\left(x \right)}$.

   Then $\operatorname{du}{\left(x \right)} = 1$.

   To find $v{\left(x \right)}$:

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

   Now evaluate the sub-integral.

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

   $$\int \left(- \cot{\left(x \right)}\right)\, dx = - \int \cot{\left(x \right)}\, dx$$

   1. Rewrite the integrand:

      $$\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$$

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

      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$:

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

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

      Now substitute $u$ back in:

      $$\log{\left(\sin{\left(x \right)} \right)}$$

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

3. Add the constant of integration:

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

The answer is: