Integral of (1-csc(t)cot(t))dt dx

1. Integrate term-by-term:

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

      $$\int \left(- \cot{\left(t \right)} \csc{\left(t \right)}\right)\, dt = - \int \cot{\left(t \right)} \csc{\left(t \right)}\, dt$$

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

         $$\int \cot{\left(t \right)} \csc{\left(t \right)}\, dt = - \int \left(- \cot{\left(t \right)} \csc{\left(t \right)}\right)\, dt$$

         1. The integral of cosecant times cotangent is cosecant:

            $$\int \left(- \cot{\left(t \right)} \csc{\left(t \right)}\right)\, dt = \csc{\left(t \right)}$$

         So, the result is: $- \csc{\left(t \right)}$

      So, the result is: $\csc{\left(t \right)}$

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

      $$\int 1\, dt = t$$

   The result is: $t + \csc{\left(t \right)}$

2. Add the constant of integration:

   $$t + \csc{\left(t \right)}+ \mathrm{constant}$$

The answer is:

$$t + \csc{\left(t \right)}+ \mathrm{constant}$$