When to Use the Tabular Method

Use tabular integration when you have ∫ p(x)·f(x) dx where p(x) is a polynomial that will eventually become zero when differentiated repeatedly. Examples: ∫x³sin(x)dx, ∫x⁴eˣdx, ∫x²cos(3x)dx.

How to Set Up the Table

📋 Table Structure
Column 1Sign: +, −, +, −, +, ... alternating starting with +
Column 2u and its successive DERIVATIVES until you reach 0
Column 3dv and its successive ANTIDERIVATIVES (same number of rows)
MultiplyMultiply diagonally: sign × derivative × antiderivative of the row below, then add all products

Example 1 — ∫ x²eˣ dx

📋 Full tabular setup
Setupu column: x², 2x, 2, 0 | dv column: eˣ, eˣ, eˣ, eˣ
Signs+, −, +
Products(+)x²·eˣ + (−)2x·eˣ + (+)2·eˣ
Answereˣ(x² − 2x + 2) + C

Example 2 — ∫ x³sin(x) dx

📋 Tabular setup
u columnx³ → 3x² → 6x → 6 → 0
dv columnsin x → −cos x → −sin x → cos x → sin x
Signs+, −, +, −
Products(+)x³(−cos x) + (−)3x²(−sin x) + (+)6x(cos x) + (−)6(sin x)
Answer−x³cos x + 3x²sin x + 6x cos x − 6 sin x + C

Example 3 — ∫ x²cos(2x) dx

📋 Tabular setup
u columnx² → 2x → 2 → 0
dv columncos(2x) → sin(2x)/2 → −cos(2x)/4 → −sin(2x)/8
Products(+)x²·sin(2x)/2 + (−)2x·(−cos(2x)/4) + (+)2·(−sin(2x)/8)
Answerx²sin(2x)/2 + x cos(2x)/2 − sin(2x)/4 + C

Example 4 — ∫ x⁴eˣ dx (4 applications)

📋 Tabular setup
u columnx⁴ → 4x³ → 12x² → 24x → 24 → 0
dv columneˣ → eˣ → eˣ → eˣ → eˣ → eˣ
Answereˣ(x⁴ − 4x³ + 12x² − 24x + 24) + C

When to Stop the Table

Stop when the u column reaches 0 (polynomial fully differentiated). The final row is not included in the products — the table stops at zero. You always take one fewer product than the number of rows in the u column (not counting the zero).

⚠ Alternating Signs

The signs strictly alternate +, −, +, −, ... starting with +. A common error is resetting the sign pattern mid-table. Each diagonal product gets the sign of the row where the derivative lives, not the antiderivative.

Frequently Asked Questions

Can I use tabular for ∫ ln(x)·x dx?▾
Yes — put u = ln(x) (differentiates to 1/x, then becomes messy) or better, put u = x (polynomial, reaches 0 fast). Let u = x, dv = ln(x)dx — but this requires integrating ln(x) which you would need to know. Usually for ln(x)·xⁿ, LIATE gives u = ln(x), dv = xⁿ dx, and one application of IBP is enough without needing the tabular method.
Does the tabular method work if neither function reaches zero?▾
No — the standard tabular method only works when the u column eventually reaches zero. For integrals like ∫eˣsin(x)dx where neither function terminates, you need the IBP loop method instead: apply IBP twice and solve algebraically for the original integral.

Why the Table Is Faster

The standard IBP formula applied twice for ∫x²eˣdx requires writing the formula out in full both times, carrying intermediate terms forward, and combining at the end — about 12 lines of algebra. The tabular method does the same computation in a 4-row table with 3 multiplication arrows. For ∫x⁴eˣdx (four applications), the saving is even greater.

The IBP Loop — When Tables Don't Work

For ∫eˣsin(x)dx, neither function reduces to zero. The tabular method would produce an infinite table. Instead, apply IBP twice and use algebra:

📋 ∫eˣsin(x)dx — the loop technique
First IBPu=sin x, dv=eˣdx → eˣsin(x) − ∫eˣcos(x)dx
Second IBPu=cos x, dv=eˣdx → eˣcos(x) + ∫eˣsin(x)dx
CombinedI = eˣsin(x) − [eˣcos(x) + I] = eˣsin(x) − eˣcos(x) − I
Solve for I2I = eˣ(sin x − cos x) → I = eˣ(sin x − cos x)/2 + C

When the Tabular Method Applies

📋 Decision table
✅ Use tabular∫xⁿ·eˣdx, ∫xⁿ·sin(x)dx, ∫xⁿ·cos(x)dx — polynomial eventually hits zero
❌ Use loop∫eˣ·sin(x)dx, ∫eˣ·cos(x)dx — neither function terminates
⚠ Use LIATE∫ln(x)·xⁿdx — one application of IBP suffices, tabular is overkill

Practice with Full Answers

📋 ∫x³cos(x)dx
Derivatives of x³x³ → 3x² → 6x → 6 → 0
Integrals of cos(x)cos(x) → sin(x) → −cos(x) → −sin(x) → cos(x)
Signs: +−+−x³sin(x) + 3x²cos(x) − 6x·sin(x) − 6cos(x) + C
AM
Dr. Aisha Malik, PhD Mathematics
Senior Lecturer in Applied Mathematics · 12 years teaching calculus
Dr. Malik holds a PhD in Applied Mathematics from the University of Edinburgh and has taught calculus to over 4,000 students. She has reviewed this article for mathematical accuracy and pedagogical clarity.
Technically reviewed by: Prof. James Chen, Stanford Mathematics Department · April 2026