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University Math Lab: Systems & Structures FREE
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Integration by Parts
Learning Objectives
- Apply integration by parts
- Choose u and dv strategically
- Handle repeated IBP
Key Concepts
IBP Formula
∫u dv = uv - ∫v du
LIATE Rule
Priority for u: Log, Inverse trig, Algebraic, Trig, Exponential
Theory
**∫u dv = uv - ∫v du**
**LIATE Rule** for choosing u (first match wins):
1. **L**ogarithmic (ln x)
2. **I**nverse trig (arctan x)
3. **A**lgebraic (x², x)
4. **T**rigonometric (sin x, cos x)
5. **E**xponential (eˣ)
**Tabular method:** For repeated IBP (e.g., ∫x³eˣ dx), use tabular integration.
**Cyclic IBP:** ∫eˣsinx dx requires IBP twice, then solve algebraically.
Examples
∫x·eˣ dx
Solution: u=x, dv=eˣdx. = xeˣ - ∫eˣdx = xeˣ - eˣ + C = eˣ(x-1) + C