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Add u-substitution rule (f'(x) * g(f(x))) #7

Description

@Pooria-Lari

Goal

Implement a basic u-substitution integrator rule to handle patterns like:

  • ∫ 2xcos(x^2) dx = sin(x^2)
  • ∫ 3exp(3x) dx = exp(3*x)
  • ∫ (2*x)/(x^2+1) dx = ln(x^2+1)

Scope (v0.4)

  • Only handle multiplication forms: f'(x) * g(f(x))
  • Support g(u) as:
    • sin(u), cos(u), exp(u)
    • 1/u (to produce ln(u))
  • No full general Risch algorithm, no integration by parts yet.

Approach

  • Use existing diff() + simplify() as a checker:
    Try candidates u inside factors and verify if one factor equals diff(u).
  • Extend integrate.py with a new rule before raising IntegrateError.

Acceptance Criteria

  • New tests in tests/test_integrate_subst.py (or extend test_integrate.py) pass:
    • roundtrip: simplify(diff(integrate(f))) == simplify(f)
    • for the listed examples above
  • Existing tests remain green.

Notes

  • Keep implementation conservative: better to reject unknown forms than return wrong answers.

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