Integration of functions of f(ax+b) AASL 5.10
Keywords
[table][br][br][tr][br][td]Integration[/td][br][td]積分[/td][br][td]적분[/td][br][td]积分[/td][br][/tr][br][tr][br][td]Substitution[/td][br][td]置換[/td][br][td]치환[/td][br][td]代换[/td][br][/tr][br][tr][br][td]Integral[/td][br][td]積分式[/td][br][td]적분식[/td][br][td]积分式[/td][br][/tr][br][tr][br][td]Function[/td][br][td]関数[/td][br][td]함수[/td][br][td]函数[/td][br][/tr][br][tr][br][td]Sine[/td][br][td]サイン[/td][br][td]사인[/td][br][td]正弦[/td][br][/tr][br][tr][br][td]Cosine[/td][br][td]コサイン[/td][br][td]코사인[/td][br][td]余弦[/td][br][/tr][br][tr][br][td]Constant of integration[/td][br][td]積分定数[/td][br][td]적분 상수[/td][br][td]积分常数[/td][br][/tr][br][tr][br][td]Efficiency[/td][br][td]効率[/td][br][td]효율성[/td][br][td]效率[/td][br][/tr][br][tr][br][td]Mathematical theory[/td][br][td]数学理論[/td][br][td]수학 이론[/td][br][td]数学理论[/td][br][/tr][br][tr][br][td]Computational tools[/td][br][td]計算ツール[/td][br][td]계산 도구[/td][br][td]计算工具[/td][br][/tr][br][/table][br]
[table][br][tr][br][td][b]Factual Questions[/b][/td][br][td][b]Conceptual Questions[/b][/td][br][td][b]Debatable Questions[/b][/td][br][/tr][br][tr][br][td]1. What is integration by substitution, and how is it performed?[/td][br][td]1. How does the concept of u-substitution simplify the process of finding integrals?[/td][br][td]1. Is the technique of integration by substitution always the most efficient method for solving integrals of the form f(ax+b)?[/td][br][/tr][br][tr][br][td]2. What is the integral of sin(x) with respect to x?[/td][br][td]2. Why does integration by substitution work, and what does it tell us about the relationship between functions?[/td][br][td]2. In what ways might the process of integration by substitution evolve with the advancement of mathematical theory?[/td][br][/tr][br][tr][br][td]3. What is the role of the constant of integration, often denoted as 'c', in indefinite integrals?[/td][br][td]3. How do the properties of sine and cosine functions affect their integrals?[/td][br][td]3. Should the mastery of manual integration techniques be a priority in a world with increasing reliance on computational tools?[/td][br][/tr][br][/table][br]
How does the integral [math]\int(ax+b)^{-1}dx[/math] differ from the integral of [math](ax+b)^n[/math] for [math]n\ne-1[/math]?[br]
2. What is the general form of the antiderivative for [math]\int(ax+b)^ndx[/math] when [math]n\ne-1[/math]?[br][br]
How is the antiderivative of [math]\int\frac{1}{ax+b}dx[/math] uniquely defined compared to other powers of [math](ax+b)[/math]?
Lesson plan - Mastering Integration by Substitution
