Back to Integration and Accumulation of Change

Lesson PreviewLesson 12 of 14 in this unit

6.12 Integrating Using Integration by Parts

Part of Integration and Accumulation of Change in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

Start free
This topic is BC only. Integration by parts is based on reversing the product rule. Since the product rule says (uv)'=u'v+uv', integration by parts rearranges that idea into the formula
∫u dv=uv−∫v du\int u\,dv = uv-\int v\,du
This method is useful when an integrand is a product of two functions and one of them becomes simpler when differentiated. Common choices include products involving polynomials, logarithms, inverse trig functions, and exponentials or trig functions.

The key to success is choosing u and dv well. Usually you want u to be a function that becomes simpler after differentiation, while dv should be something whose antiderivative is manageable. After choosing u and dv, you compute du and v, plug into the formula, and simplify. You should think of integration by parts as a strategic rewriting tool: the first integral is hard, but after the substitution into the formula, the new integral is easier.

In this unit

14 lessons

  1. 6.1Exploring Accumulations of ChangePreview lesson
  2. 6.2Approximating Areas with Riemann SumsPreview lesson
  3. 6.3Riemann Sums, Summation Notation, and Definite Integral NotationPreview lesson
  4. 6.4The Fundamental Theorem of Calculus and Accumulation FunctionsPreview lesson
  5. 6.5Interpreting the Behavior of Accumulation Functions Involving AreaPreview lesson
  6. 6.6Applying Properties of Definite IntegralsPreview lesson
  7. 6.7The Fundamental Theorem of Calculus and Definite IntegralsPreview lesson
  8. 6.8Finding Antiderivatives and Indefinite Integrals: Basic Rules and NotationPreview lesson
  9. 6.9Integrating Using SubstitutionPreview lesson
  10. 6.10Integrating Functions Using Long Division and Completing the SquarePreview lesson
  11. 6.11Selecting Techniques for AntidifferentiationPreview lesson
  12. 6.12Integrating Using Integration by PartsThis lesson
  13. 6.13Integrating Using Linear Partial FractionsPreview lesson
  14. 6.14Evaluating Improper IntegralsPreview lesson

Continue the full lesson in Stellar

Create a free account to unlock the remaining sections, checkpoints, course practice, and progress tracking for AP® Calculus BC.

Start free