Now let's try to find the anti-derivative of \( \csc(x)\):
\[\int \csc(x) \ dx \]
Let's multiply the the numerator and denominator by \( \csc(x) + \cot(x)\):
\[\int \csc(x) \ dx = \int \frac{ \csc^2(x) + \csc(x)\cot(x)}{ \csc(x) + \cot(x)} \ dx \]
Let \(u = \csc(x)+ \cot(x) \), and that means \(du = -\csc^2(x) - \csc(x)\cot(x) \ dx\):
\[\int \frac{-\csc^2(x) - \csc(x)\cot(x)}{ \csc(x) + \cot(x)} \ dx = - \int \frac{\csc^2(x) + \csc(x)\cot(x)}{ \csc(x) + \cot(x)} \ dx = - \int \frac{1}{u} \ du\]
If we evaluate the integral:
\[- \ln|u| + C = - \ln\left(\left| \csc(x) + \cot(x) \right|\right) +C\]
We will also derive another antiderivative for \( \csc(x)\). Let's multiply the the numerator and denominator by \( \csc(x) - \cot(x)\):
\[\int \csc(x) \ dx = \int \frac{ \csc(x) (\csc(x) - \cot(x))}{ \csc(x) - \cot(x)} \ dx \]
Let \(u = \csc(x)- \cot(x) \), and that means \(du = - \csc(x)\cot(x) + \csc^2(x) \ dx\):
\[\int \csc(x) \ dx = \int \frac{ \csc(x) (\csc(x) - \cot(x))}{ \csc(x) - \cot(x)} \ dx = \int \frac{1}{u} \ du\]
If we evaluate the integral:
\[\ln|u| + C = \ln\left(| \csc(x) - \cot(x) |\right) +C\]