Integration by Parts
Product rule for integration
About Integration by Parts
The Integration by Parts represents product rule for integration. This calculus formula is fundamental to mathematical analysis and serves as a cornerstone concept that students and professionals encounter throughout their mathematical journey. Its importance extends beyond pure mathematics into applied fields where quantitative analysis is required.
This formula is essential in Calculus and Mathematical analysis. It serves as a building block for more advanced mathematical theory and provides the foundation needed to understand complex mathematical relationships. Whether you're studying mathematics, physics, engineering, or economics, familiarity with this formula enhances your analytical capabilities.
Practical applications of the Integration by Parts include Calculus problems, Physics calculations, Engineering analysis, among others. Understanding and correctly applying this formula enables problem-solvers to approach challenges more systematically and efficiently. Mastery of this concept not only expands your mathematical knowledge but also improves your overall quantitative reasoning skills.
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LaTeX Code
\int u \, dv = uv - \int v \, du
Formula Information
Difficulty Level
Prerequisites
Discovered
17th century
Discoverer
Brook Taylor
Real-World Applications
Examples
Mathematical Fields
Keywords
Related Topics
Important Notes
Integration by parts is the integration equivalent of the product rule for differentiation. Choose u and dv carefully.
Alternative Names
Common Usage
Formula Variations
Frequently Asked Questions
When should I use integration by parts?
Use integration by parts when you have an integral of a product of two functions where one function becomes simpler when differentiated and the other becomes manageable when integrated. Common cases: ∫x·eˣdx, ∫x·sin(x)dx, ∫ln(x)dx, ∫x·cos(x)dx. The goal is to simplify the integral.
How do I choose u and dv?
Use the LIATE rule as a guide: Logarithmic, Inverse trigonometric, Algebraic (polynomials), Trigonometric, Exponential. Choose u from the category that appears first. For example, in ∫x·eˣdx, x is Algebraic (u), eˣ is Exponential (dv). The goal is to make ∫v du simpler than the original integral.
What is the LIATE rule?
LIATE is a mnemonic for choosing u in integration by parts: L = Logarithmic (ln x, log x), I = Inverse trigonometric (arcsin, arctan), A = Algebraic (polynomials, xⁿ), T = Trigonometric (sin, cos), E = Exponential (eˣ, aˣ). Choose u from the category that appears first in this order.
What if integration by parts doesn't simplify the integral?
If the new integral ∫v du is just as difficult or more difficult, you may have chosen u and dv incorrectly. Try swapping them. Sometimes you need to apply integration by parts twice (like ∫eˣsin(x)dx), or combine it with other techniques. In some cases, the integral might require a different method entirely.
Can I use integration by parts more than once?
Yes! Sometimes you need to apply it multiple times. For example, ∫x²eˣdx requires applying it twice. After the first application, you get a simpler integral that may still need integration by parts. Continue until you reach an integral you can solve directly.
What's the tabular method?
The tabular method is a shortcut for repeated integration by parts. Create a table: alternate signs in one column, derivatives of u in another, integrals of dv in a third. When a derivative becomes 0 or repeats, you can read off the answer. This is especially useful for integrals like ∫xⁿeˣdx or ∫xⁿsin(x)dx.
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Quick Details
- Category
- Calculus
- Difficulty
- Intermediate
- Discovered
- 17th century
- Discoverer
- Brook Taylor
- Formula ID
- integration-parts