Overview
This lecture introduces the integration by parts technique, derives its formula, and demonstrates its use through multiple examples, including definite and indefinite integrals.
Introduction to Integration by Parts
- Some integrals cannot be solved by substitution alone and require other techniques.
- Integration by parts is related to the product rule for derivatives.
Derivation and Formula
- Start with the product rule for derivatives: (f·g)' = f'g + fg'.
- Integrating both sides gives: ∫f' g dx = f g - ∫f g' dx.
- Using substitutions u = f(x), dv = g'(x) dx, the standard formula is:
- ∫u dv = u v - ∫v du.
Using Integration by Parts
- Choose u and dv so that du and v are easy to compute and ∫v du is simpler than the original integral.
- If the integral after applying the formula is still complex, try different choices for u and dv.
Examples and Applications
- For ∫x e^(6x) dx, let u = x, dv = e^(6x) dx; compute accordingly.
- For definite integrals, apply the formula: ∫ₐᵇ u dv = [u v]ₐᵇ - ∫ₐᵇ v du.
- Some problems require more than one application of integration by parts.
- Not all integration by parts problems follow a strict pattern (e.g., ∫ln(x) dx).
Special Cases and Tips
- If both choices for u and dv seem difficult, look for substitutions or rearrangements.
- Some integrals may loop back to the original integral; solve algebraically when this occurs.
- The tabular (diagonal) method speeds up integration by parts when one factor repeatedly differentiates to zero.
Key Terms & Definitions
- Integration by Parts — An integration technique based on the product rule, used to integrate products of functions.
- u — Part of the integrand chosen for easy differentiation.
- dv — Remaining part of the integrand, chosen for easy integration.
- Tabular Method — A shortcut for multiple applications of integration by parts using a table and alternating signs.
Action Items / Next Steps
- Practice integration by parts with provided problem sets.
- Review standard substitutions and the product rule for derivatives.
- Memorize the integration by parts formula and practice identifying good choices for u and dv.