Lecture on Integration by Parts

Introduction

• Topic: Integration by Parts
• Relation: Connected to the product rule from Calculus I (Cal I)

Indefinite Integral

• Definition: A family of functions.
  • Notation: ∫f(x)dx
  • Form: Big F(x) + C, where F'(x) = f(x)
  • C: Represents any constant.
  • Example: ∫2x^5 dx = (1/3)x^6 + C

Definite Integral

• Definition: A number that represents area under the curve.
  • Notation: ∫ from a to b f(x) dx
  • Riemann Sum: Limit of sums of function values multiplied by interval widths.
  • Area Interpretation: Represents exact area under the curve if f is positive.
  • Example: ∫ from 0 to 1 of 2x^6 dx = 1/3(1^6 - 0^6)

Purpose of Integration

• Applications:
  • Physics: Relationship between velocity and position.
  • Practical Uses: Calculating quantities like water in a cylinder.

Fundamental Theorem of Calculus

• Relationship: Connects indefinite and definite integrals.
  • Statement: If F is an antiderivative of f, then ∫ from a to b f(x) dx = F(b) - F(a).
  • Implication: Simplifies calculation of definite integrals.

Integration by Parts

• Origin: Derives from the product rule.
• Product Rule: For differentiable u and v, (uv)' = u'v + uv'.
• Integration by Parts Formula:
  • Definite: ∫ from a to b u(x)v'(x)dx = [u(x)v(x)] from a to b - ∫ from a to b v(x)du
  • Indefinite: ∫ u dv = uv - ∫ v du

Notation

• Shorthand:
  • u = U(x), dv = v'(x)dx
  • Expression: ∫ u dv = u*v - ∫ v du

Examples and Strategy

• Goal: Convert difficult integrals into easier ones by strategic choice of u and dv.

  • Criteria:
    • Choose u such that its derivative is simpler.
    • Choose dv such that its integral is manageable.

• Example Strategy: If u = x^2, then du simplifies to 2x, making integration more manageable.

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• Further examples and practice are crucial for mastering integration by parts.
• Importance of understanding how to choose u and dv for simplification.