The LIATE Rule

Definition

The LIATE Rule is a heuristic (a rule of thumb) used in calculus to help choose which part of an integrand should be assigned to the variable $u$ when performing Integration by Parts.

Why It Matters

Integration shouldn’t be a guessing game. The LIATE rule provides the strategic hierarchy needed to pick the right components for integration by parts, turning a potentially circular problem into a straightforward, solvable task.

Core Concepts

The acronym LIATE stands for the preference order for selecting $u$:

1. L: Logarithmic functions (e.g., $\ln x, \log_a x$)

   • How to read: “The natural log of x or the log base a of x.”
   • Meaning: Logs differentiate to simpler algebraic forms—top priority for $u$.

2. I: Inverse Trigonometric functions (e.g., $\arctan x, \arcsin x$)

   • How to read: “The arc tangent of x or the arc sine of x.”
   • Meaning: Inverse trig also simplifies under differentiation.

3. A: Algebraic functions (e.g., $x^n, 3x^2, \sqrt{x}$)

   • How to read: “The quantity x to the n, three x squared, or the square root of x.”
   • Meaning: Polynomials reduce degree when differentiated—good $u$ candidates.

4. T: Trigonometric functions (e.g., $\sin x, \cos x$)

   • How to read: “The sine of x or the cosine of x.”
   • Meaning: Trig cycles under differentiation—often better as $dv$ than $u$.

5. E: Exponential functions (e.g., $e^x, 2^x$)

   • How to read: “The quantity e to the x or two to the x.”
   • Meaning: Exponentials stay exponential under differentiation—usually assign to $dv$.

Connected Concepts

• Integration by Parts
• Inversion
• Basic Differentiation Rules
• Inverse Functions
• Transcendental Functions
• Fundamental Theorem of calculus
• First Principles Thinking