Integration by Parts
Updated 2026-08-10
INTRODUCTION
English translation pending.
CORE DEFINITION
Integration by parts is a technique derived from the product rule for differentiation. It rewrites an integral of the form u times dv as uv minus the integral of v du, transforming a difficult integral into a hopefully easier one, and it can be applied repeatedly when the new integral still resists direct solution. The LIATE heuristic guides the choice of u by ranking logarithmic, inverse trigonometric, algebraic, trigonometric, and exponential functions.
SCAFFOLDING EFFECT
Reduce cognitive load
- Choose u wisely: pick u by the LIATE order so the remaining integral gets simpler. - Compute the pieces: differentiate u and integrate dv to obtain du and v. - Apply the formula: substitute into uv minus the integral of v du.
Anchor fast decisions
The formula is the product rule read backwards, so it moves the difficult part of the integrand from one factor to another. Differentiating u reduces the algebraic or logarithmic complexity while integrating dv raises it elsewhere, so a wise choice shifts difficulty toward a term that eventually cancels or repeats.
MINIMUM ACTION
In progress 0/1Practice this model in one real situation:
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Source support: Explicit
- en.wikipedia.orghttps://en.wikipedia.org/wiki/Integration_by_partsverified
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