Question

How do you integrate `int x^2 csc ^2 x^2 dx` using integration by parts?

Answer 1

I don't believe that you do. Wolfram Alpha gives "no result found in terms of standard mathematics functions"

Explanation:

Perhaps if the integrand were `x^3csc^2(x^2)`

Clearly choosing `u = csc^2(x^2)` and `dv = x^2 dx` will not result in anything we can integrate.
(We get `vdu` involving `x^4csc^2(x^2) cot(x^2)`, which is not promising.)

We can integrate `x csc^2(x^2)` by substitution, but our `vdu` involves just `kint cot(x^2)dx`, which does not look like something we can do.