Integration by Parts Formula
Integration by parts converts a hard integral into a simpler one.
Learn the LIATE rule, formula derivation, and worked examples with polynomials and trig.
The Formula
Integration by parts is a technique for integrating a product of two functions. It is derived from the product rule of differentiation and essentially reverses it. When you encounter an integral that is a product of two different types of functions — such as a polynomial times a trigonometric function — integration by parts is often the right tool.
Variables
| Symbol | Meaning | Notes |
|---|---|---|
| u | First function (you differentiate this) | Choose using LIATE |
| dv | Second part (you integrate this) | Must be integrable |
| du | Derivative of u (du = u′ dx) | Computed after choosing u |
| v | Antiderivative of dv (v = ∫ dv) | Computed after choosing dv |
The LIATE Rule — Choosing u
Choose u as the function that appears first in this priority order:
- Logarithmic functions: ln(x), log(x)
- Inverse trig: arcsin(x), arctan(x)
- Algebraic: xⁿ, polynomials
- Trigonometric: sin(x), cos(x)
- Exponential: eˣ, aˣ
The remaining factor becomes dv. This rule works in most cases but is a guideline, not an absolute law.
Example 1 — Polynomial × Exponential
∫ x eˣ dx
Choose u = x (Algebraic), dv = eˣ dx (Exponential)
du = dx, v = eˣ
∫ x eˣ dx = x·eˣ − ∫ eˣ dx
= x·eˣ − eˣ + C = eˣ(x − 1) + C
Example 2 — Polynomial × Logarithm
∫ x ln(x) dx
Choose u = ln(x) (Logarithmic), dv = x dx (Algebraic)
du = (1/x) dx, v = x²/2
∫ x ln(x) dx = (x²/2)·ln(x) − ∫ (x²/2)·(1/x) dx
= (x²/2)·ln(x) − ∫ x/2 dx = (x²/2)·ln(x) − x²/4 + C
= (x²/4)(2 ln(x) − 1) + C
Repeated Integration by Parts
Some integrals require multiple applications. For ∫ x² eˣ dx, apply integration by parts twice, reducing x² → x → 1. The tabular method (or "DI method") speeds this up by listing successive derivatives of u and integrals of dv in columns:
When to Use Integration by Parts
- Products of polynomials with exponentials, logarithms, or trig functions
- Integrals of ln(x), arcsin(x), arctan(x) alone — set u = the function, dv = dx
- When u-substitution doesn't simplify the integral
- Cyclic integrals (e.g., ∫ eˣ sin(x) dx) — apply twice, then solve algebraically