code: def test_dup_integrate():
      assert dup_integrate([], 1, QQ) == []
      assert dup_integrate([], 2, QQ) == []
  
      assert dup_integrate([QQ(1)], 1, QQ) == [QQ(1), QQ(0)]
      assert dup_integrate([QQ(1)], 2, QQ) == [QQ(1, 2), QQ(0), QQ(0)]
  
      assert dup_integrate([QQ(1), QQ(2), QQ(3)], 0, QQ) == [
          QQ(1),
          QQ(2),
          QQ(3),
      ]
      assert dup_integrate([QQ(1), QQ(2), QQ(3)], 1, QQ) == [
          QQ(1, 3),
          QQ(1),
          QQ(3),
          QQ(0),
      ]
      assert dup_integrate([QQ(1), QQ(2), QQ(3)], 2, QQ) == [
          QQ(1, 12),
          QQ(1, 3),
          QQ(3, 2),
          QQ(0),
          QQ(0),
      ]
      assert dup_integrate([QQ(1), QQ(2), QQ(3)], 3, QQ) == [
          QQ(1, 60),
          QQ(1, 12),
          QQ(1, 2),
          QQ(0),
          QQ(0),
          QQ(0),
      ]
  
      assert dup_integrate(
          dup_from_raw_dict({29: QQ(17)}, QQ), 3, QQ
      ) == dup_from_raw_dict({32: QQ(17, 29760)}, QQ)
  
      assert dup_integrate(
          dup_from_raw_dict({29: QQ(17), 5: QQ(1, 2)}, QQ), 3, QQ
      ) == dup_from_raw_dict({32: QQ(17, 29760), 8: QQ(1, 672)}, QQ)
  
imports: from sympy.polys.densebasic import dup_normal, dmp_normal, dup_from_raw_dict, dmp_convert, dmp_swap
  from sympy.polys.densearith import dmp_mul_ground
  from sympy.polys.densetools import dup_clear_denoms, dmp_clear_denoms, dup_integrate, dmp_integrate, dmp_integrate_in, dup_diff, dmp_diff, dmp_diff_in, dup_eval, dmp_eval, dmp_eval_in, dmp_eval_tail, dmp_diff_eval_in, dup_trunc, dmp_trunc, dmp_ground_trunc, dup_monic, dmp_ground_monic, dup_content, dmp_ground_content, dup_primitive, dmp_ground_primitive, dup_extract, dmp_ground_extract, dup_real_imag, dup_mirror, dup_scale, dup_shift, dup_transform, dup_compose, dmp_compose, dup_decompose, dmp_lift, dup_sign_variations, dup_revert, dmp_revert
  from sympy.polys.polyclasses import ANP
  from sympy.polys.polyerrors import MultivariatePolynomialError, ExactQuotientFailed, NotReversible, DomainError
  from sympy.polys.specialpolys import f_polys
  from sympy.polys.domains import FF, ZZ, QQ, EX
  from sympy.polys.rings import ring
  from sympy.core.numbers import I
  from sympy.core.singleton import S
  from sympy.functions.elementary.trigonometric import sin
  from sympy.abc import x
  from sympy.testing.pytest import raises
label_desc: Use image node_id 1 for calling a global function with example usage: test_dup_integrate() without return types