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Understanding Integration: The Antiderivative and Its Rules

This presentation covers key concepts in integration, focusing on antiderivatives and techniques for finding them. It explains the constant of integration (C), power rules, and the guessing-checking-answering method to solve integrals. The presentation also introduces the "Most Complicated Rule" for handling complex integrals and discusses the approach for definite integrals with practical examples. Integral calculations involve raising powers, using logarithmic functions, and substitution methods. Enhance your understanding of integration through this concise guide.

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Understanding Integration: The Antiderivative and Its Rules

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  1. Integration Jami Wang Period 3 EC PPT

  2. Integration – the Antiderivative • C represents Constant • Raise the power • Guess. Check. Answer. • Ex: ∫ 2X dx • Guess: X2 • Check: 2X • Answer : X2 + C • Off by a constant = good • Off by a variable = new guess

  3. Integration – the Antiderivative (continued) • When power raised = zero, use ln • Ex: ∫ 2/X dx( ∫ 2X-1dx ) • Guess: 2lnX • Check: (2) 1/x • Answer: 2lnx + Cor lnX2 + C

  4. Most Complicated Rule • Start out with the complicated part and ignore the easy part • Can’t decide the most complicated one – choose one • Ex: ∫ X (X2- 1)9 • Guess: 1/20(X2- 1)10 • Check: 10 (X2- 1)9 (2X) • Answer: 1/20 (X2- 1)10 + C

  5. Most Complicated Rule (continued) • Ex2: ∫ (ln x6 / x) dx • Guess: 1/7 (lnx)7 • Check: (lnx) 6 (1/x) • Answer: 1/7 (lnx) 7+ C

  6. Definite Integrals • Ex: 30∫ √ (y+1) dy3∫0 √ (y+1)1/2dy • Guess: 2/3 (y+1)3/2 • Check: 3/2 (y+1)1/2 • substitution then subtraction • Biggest number goes first 2/3 (y+1)3/2 30 = 2/3 (4)3/2 – 2/3 (1) 3/2*use x values to find y values • Answer: 14/3

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