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Integration by Parts: Two Applications with Symbolic Simplification

Integration by Parts: Two Applications with Symbolic Simplification
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2020-09-12 05:46
TrigonometryCalculusDerivativeIntegralGeometric Construction
This GeoGebra example demonstrates how integration by parts can transform more complicated integrals into simpler, computable forms, using symbolic manipulation tied to parameter-like quantities. It focuses on the core ideas of integration by parts, trigonometric simplification (TrigSimplify), general algebraic simplification (Simplify), and the relationship between integrals and derivatives (Derivative) for verification and intermediate steps. Construction-wise, the workflow is driven by the Integral command to build the integral expressions, followed by TrigSimplify and Simplify to standardize the results. Derivative is used to define or check derivative-related objects (e.g., quantities analogous to u′ and v′). Textfield and InputBox enable parameter entry and structured display of intermediate formulas and computed parameters (such as symbols representing integral components). Suitable for high school advanced learners and early university calculus courses, especially for teaching and practicing integration by parts and symbolic verification.

GeoGebra Commands & Parameters

  1. Integral(v_1') → v_1
  2. TrigSimplify(u_1'(x)) → u_{1}'
  3. TrigSimplify((u_{1}'(x) * v_1(x))) → D
  4. Simplify(D(x) / ((u(x) * v'(x)))) → q_1
  5. Textfield(u) → InputBox1
  6. Integral(v') → v
  7. Derivative(u) → u'
  8. Textfield(v') → InputBox2
  9. Simplify((u'(x) * v(x))) → w
  10. Integral(w) → a_1
  11. TrigSimplify((u(x) * v(x)) - Integral[(u'(x) * v(x))]) → J
  12. Simplify(Integral[v']) → k
  13. Textfield(d) → InputBox3
  14. Textfield(c) → InputBox4
  15. TrigSimplify((-q_1(x))) → K
  16. TrigSimplify(Integral[(u(x) * v'(x))]) → L
  17. Simplify((u(x) * v(x))) → A
  18. TrigSimplify((u'(x) * v(x))) → B
  19. TrigSimplify((u_1(x) * v_1(x))) → C
  20. Textfield(u_1) → InputBox5
  21. Textfield(v_1') → InputBox6
  22. TrigSimplify(A(x) - C(x) + K(x)) → I_1
  23. Simplify(1 - q_1(x)) → q_2
  24. Simplify(A(x) - C(x)) → E
  25. Simplify(E(x) / q_2(x)) → G
  26. Simplify(A(x) - C(x)) → F

AI Drawing Prompt Reference

Create a symbolic diagram for integration by parts showing connected intermediate expressions depending on parameters. Present functions u and v together with their derivatives u' and v' and indicate product terms such as u'·v, u·v, and the resulting integrands w and a_1. Include the derived quantities obtained by integrating these products, especially an expression built from (u·v) minus ∫(u'·v), labeled J, and the simplified form of ∫v' labeled k. Display the parameter-style ratio q_1 and its simplification to q_2, and use it to form a quotient that produces G (using the same A−C relationship). Mirror the construction with u1 and v1' (and their products u1·v1, u1'·v1) and show the combined identity expressions such as I_1, then connect related simplified forms E, F, and K/ L through their defined relationships.
The prompt above was compiled from the original drawing commands and is for reference only.