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LIATE rule — Definition, Formula & Examples

The LIATE rule is a mnemonic that helps you decide which part of an integrand to choose as uu when applying integration by parts. The letters stand for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential — pick uu from whichever category appears earliest in this list.

LIATE is a heuristic ordering of function types — Logarithmic (e.g., ln⁡x\ln x), Inverse trigonometric (e.g., arctan⁡x\arctan x), Algebraic (e.g., x2x^2), Trigonometric (e.g., sin⁡x\sin x), Exponential (e.g., exe^x) — used to assign uu in the integration by parts formula ∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du. The function whose type appears earliest in the LIATE sequence is selected as uu, and the remainder of the integrand is assigned to dvdv.

How It Works

When you face an integral with two different types of functions multiplied together, scan the LIATE list from left to right. The first function type you encounter becomes uu, while everything else becomes dvdv. For example, if your integrand contains both an algebraic term (xx) and an exponential term (exe^x), algebraic comes before exponential in LIATE, so you set u=xu = x. This ordering works because functions earlier in the list typically become simpler when differentiated, which is exactly what happens to uu during integration by parts. The rule is a guideline, not a guarantee — occasionally you may need to deviate from it.

Worked Example

Problem: Evaluate ∫x ex dx\int x\, e^x\, dx using the LIATE rule to choose uu.
Apply LIATE: The integrand has an algebraic function (xx) and an exponential function (exe^x). Algebraic (A) comes before Exponential (E) in LIATE, so set u=xu = x and dv=ex dxdv = e^x\,dx.
u=x,dv=ex dxu = x,\quad dv = e^x\,dx
Differentiate and integrate: Compute dudu and vv.
du=dx,v=exdu = dx,\quad v = e^x
Apply integration by parts: Substitute into ∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du.
∫x ex dx=xex−∫ex dx=xex−ex+C\int x\,e^x\,dx = x e^x - \int e^x\,dx = x e^x - e^x + C
Answer: xex−ex+Cx e^x - e^x + C

Why It Matters

Choosing uu poorly in integration by parts can make an integral harder instead of easier. The LIATE rule gives you a reliable starting point in Calculus II, where integration by parts appears repeatedly in problems involving mixed function types.

Common Mistakes

Mistake: Treating LIATE as an absolute rule and never deviating from it.
Correction: LIATE is a heuristic. Some integrals, such as ∫exsin⁡x dx\int e^x \sin x\,dx, require applying integration by parts twice regardless of which factor you call uu. Always verify that your choice of uu actually simplifies the resulting integral.

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