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Integration by parts twice: Basic Calculus Example

Integration by parts twice: Basic Calculus Example
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2020-10-19 17:51
CalculusDerivativeIntegralGeometric Construction
This example demonstrates the application of integration twice, focusing on calculus concepts such as the Fundamental Theorem of Calculus and integration techniques. Construction involves GeoGebra commands like Integral, TrigSimplify, Simplify, and Derivative to handle trigonometric and polynomial expressions. Suitable for high school students learning calculus fundamentals or for mathematical teaching demonstrations.

GeoGebra Commands & Parameters

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

AI Drawing Prompt Reference

Draw the integral graph of the product of functions u(x) and v(x), as well as their derivatives u'(x) and v'(x). First, input u(x) and v'(x), then calculate the integral of the product of u(x) and v(x). Next, input u'(x) and v(x), and calculate the integral of their product. Finally, use integration by parts to calculate the integral of the product of u(x) and v'(x), and the integral of the product of u'(x) and v(x).
The prompt above was compiled from the original drawing commands and is for reference only.