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What is the rule of UV?

Published in Integration Techniques 3 mins read

The "rule of UV" refers to a specific application of integration by parts, which is a technique used to integrate the product of two functions.

Understanding Integration by Parts with the UV Formula

The core idea behind the integration by parts method, especially when dealing with functions in the form of a product (UV), is to break down a complex integral into simpler, more manageable parts. This formula is essential when other basic integration rules fail.

The UV Formula

The integration of uv formula is given as:
∫ uv dx = u ∫ v dx - ∫ (u' ∫ v dx) dx

Where:

  • u is one function of x
  • v is another function of x
  • u' represents the derivative of u with respect to x
  • ∫ v dx is the integral of v with respect to x


How the Formula Works:

  1. Identify u and v: The first step is to choose which function will be designated as u and which will be v. This selection isn't arbitrary; it's often based on which choice will simplify the resulting integral. A common guide for choosing u is the "LIATE" rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential).

  2. Calculate Derivatives and Integrals: Once u and v are chosen, we need to calculate the derivative of u (denoted as u') and the integral of v (denoted as ∫ v dx).

  3. Apply the Formula: Next, we apply these elements to the integration by parts formula: ∫ uv dx = u ∫ v dx - ∫ (u' ∫ v dx) dx

  4. Evaluate the Resulting Integrals: The goal is that ∫ (u' ∫ v dx) dx is a more manageable integral to solve than the initial ∫ uv dx. After integration, add the constant of integration.

Practical Example:

Let’s say you want to integrate ∫ x * sin(x) dx

*   **Choose u and v**: Here `u = x` (algebraic) and `v = sin(x)` (trigonometric)
*   **Calculate u' and ∫v dx:** 
    *   `u' = 1`
    *   `∫v dx = -cos(x)`
*   **Apply the formula:**
    `∫ x * sin(x) dx = x * (-cos(x)) - ∫ 1 * (-cos(x)) dx`
*   **Evaluate the resulting integral:** 
    `= -x*cos(x) + ∫ cos(x) dx`
    `= -x*cos(x) + sin(x) + C`


When to Use the UV Rule

This integration by parts method is particularly useful when:

  • The integrand is the product of two different types of functions (e.g., algebraic and trigonometric, exponential and logarithmic).
  • Direct integration is not possible or practical.
  • The integration of one of the functions can be easily obtained.


Key Points to Remember:

  • Choosing the right u and v can significantly simplify the problem.
  • The formula is used to convert one integral into another, potentially easier integral.
  • The integration of UV is a powerful technique that is very common in calculus.