Integration by Parts Using the Tabular Method

The tabular method, also known as the “method of integration by parts,” is an efficient way to integrate products of functions when repeated integration by parts is required. This method is particularly useful for functions that involve polynomial and exponential or trigonometric terms.

Contents

What is Tabular method for integration by parts?

Integration by parts tabular method (also called the DI method of Integration by parts) is a short method for integration to solve the integral problem quickly, instead of using the lengthy and tedious process of integration by parts traditional method.

The advantage of the tabular integration by parts method is that it can save huge time in solving the problem. It gives the solution fairly accurate than the integration by parts method. [1]

Tabular Method Step-by-Step

Here is the step-by-step process to set up a DI table for integration by parts.

1. Select Your Terms (LIATE Rule)

Split your integral ∫ u dv into two parts. Choose one part to differentiate (D) and one part to integrate (I).

Use the LIATE acronym to choose your D term: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential.
The category closest to the top of LIATE becomes your D term. The rest becomes your I term.

2. Set Up the Three Columns

Draw a table with three distinct columns from left to right:

Sign Column: Always start with a plus sign (+) in the first row. Alternate signs (–, +, –) for each row downwards.
D Column: Place your chosen D term in the first row.
I Column: Place your chosen I term (excluding dx) in the first row.

3. Fill the Table Downward

Differentiate Downward: Take the derivative of the D term row by row. If it is a polynomial (like x²), keep going until you reach 0.
Integrate Downward: Take the integral of the I term for each corresponding row. Match the number of rows you created in the D column.

4. Read the Final Answer

Diagonal Products: Multiply the terms diagonally. Multiply the first row’s Sign by the first row’s D term, then multiply that by the second row’s I term. Repeat this diagonal pattern for the next rows.
Horizontal Product (The Remainder): If you stop before reaching 0 in the D column, multiply the terms in the final row horizontally and place them inside an integral sign: ∫ (Sign) × (D) × (I) dx).
Sum It Up: Add all your diagonal products and your horizontal integral together to get your final answer.

*How to use or apply the Tabular integration by parts method and its formulas – Topblogtenz*

When can I use the Tabular integration by parts method?

The integration by parts tabular method can be applied to any function which is the product of two expressions, where one of the expressions can be differentiated until it gets zero, and another expression can be integrated simultaneously multiple times. [1]

Take a look at where we can apply the tabular integration by parts method.

When Integrand is the product of Polynomial times and something that can be repeatedly integrated. (∫ x¹⁰ · cos x dx)
Integrand multiple of power function and an exponential function. (∫ x² · e^x dx)
Integrand multiple of an exponential and trigonometry function. (∫ e^x · sin x dx)
Integrand multiple of power function and a trigonometry function. (∫ x³ · tan x dx)

Note: Tabular integration by parts method can also be used where neither of the expression differentiation goes to zero. Examples: ∫ e^2x · sin3x dx or ∫ sin3x · cos4x dx, etc. Let’s look at an example using ∫ e^2x · sin3x dx.

Because neither e^2x nor sin(3x) will ever differentiate to zero, this is a cyclic (looping) integral. We stop creating rows once the product of the final row matches a multiple of our original integral.

The DI Table Setup

We will choose D = sin(3x) and I = e^2x.

The final solution to the integral is

$\int e^{2x} \sin(3x) \, dx = \frac{e^{2x}}{13} \big(2\sin(3x) - 3\cos(3x)\big) + C$

Step-by-Step Algebraic Solution

1. Write out the equation from the table

Combine the two diagonal products and the final horizontal integral row. Let I represent our original integral ∫ e^2x · sin3x dx:

$I=\frac{1}{2}e^{2x}\sin (3x)-\frac{3}{4}e^{2x}\cos (3x)-\frac{9}{4}\int e^{2x}\sin (3x)\,dx$

2. Substitute I back into the equation

$I=\frac{1}{2}e^{2x}\sin (3x)-\frac{3}{4}e^{2x}\cos (3x)-\frac{9}{4}I$

3. Move all I terms to the left side

Add (9/4)I to both sides of the equation:

$I+\frac{9}{4}I=\frac{1}{2}e^{2x}\sin (3x)-\frac{3}{4}e^{2x}\cos (3x)$

$\frac{13}{4}I=\frac{e^{2x}}{4}\big(2\sin (3x)-3\cos (3x)\big)$

4. Isolate I

Multiply both sides by 4/13 and add the constant of integration C:

$I = \frac{e^{2x}}{13}\big(2\sin (3x)-3\cos (3x)\big) + C$

Conclusion

The tabular method or tabular integration by parts simplifies the integration process for the functions involving the products of polynomials, exponentials and trigonometric functions. By systematically creating a table to the handle the derivatives and integrals, this technique streamlines the process and reduces the complexity of the repeated integration by parts. It’s particularly useful for the integrals where traditional methods become cumbersome. The Mastering the tabular method enhances efficiency and accuracy in solving the complex integrals making it a valuable tool for both the students and professionals in calculus. [2]

References

[1] Goyal, Vishal. “Integration by Parts Tabular Method, Examples.” Topblogtenz, January 21, 2021. https://topblogtenz.com/tabular-integration-by-parts-example-formula/.

[2] “How to Integrate Using the Tabular Method.” GeeksforGeeks, August 11, 2024. https://www.geeksforgeeks.org/maths/tabular-method-integration/.

Additional Reading

“Integration by Parts.” GeeksforGeeks, January 31, 2021. https://www.geeksforgeeks.org/maths/integration-by-parts/.

Bhattacharya, Arpita. “Tabular Integration: Faster & Accurate Way to Solve Repeated Integration by Parts.” Medium, December 8, 2022. https://arpita95b.medium.com/tabular-integration-faster-accurate-way-to-solve-repeated-integration-by-parts-9cf790239c12.

Dawkins, Paul. “Calculus II.” Integration by Parts, April 28, 2025. https://tutorial.math.lamar.edu/classes/calcii/integrationbyparts.aspx.

Videos

Integration By Parts – Tabular Method

This calculus video tutorial explains how to find the indefinite integral using the tabular method of integration by parts. This video contains plenty of examples and practice problems of when you should use the tabular method and when you shouldn’t. The tabular method requires the use of 3 columns – signs, derivatives, and integrals.

DI | Tabular Method of Integration by Parts | Shortcut Method

Notes

DI

In calculus, DI (often called the DI method or tabular integration) is a shortcut for performing integration by parts. [IP]

The letters stand for:

D: Differentiate
I: Integrate

It organizes the integration by parts formula

$\int u \, dv = uv - \int v \, du$

into a clean, two-column table to easily compute integrals involving repeated differentiation, e.g.,

$\int x^2 \sin(x) \, dx$

How to Apply the Alternating Sign Rule

Start with Plus: The very first row of your table always gets a + sign.
Alternate Downward: Every row below it flips to the opposite sign (–, then +, then –, and so on).
Multiply Across the Diagonal: When reading your answer, you multiply the sign in Row n by the D term in Row n, and then by the I term in Row n + 1.

$\int u \, dv = uv - \int v \, du$

Here is how the signs align with your terms:

Sign	D (Differentiate)	I (Integrate)	How to Multiply for the Answer
+	x²	e^x	Multiply: (+) · x² · e^x
–	2x	e^x	Multiply: (-) · 2x · e^x
+	2	e^x	Multiply: (+) · 2 · e^x
–	0	e^x	(Stop here because D is 0)

Combining the terms gives the final answer:

$x^{2}e^{x}-2xe^{x}+2e^{x}+C$

Constant of Integration

The Constant of Integration (+C) is a placeholder for a fixed number that we add to the end of an integral.

Derivatives Destroy Numbers

When you take the derivative of a function, you are finding its speed or slope. The derivative of any regular number (like 5, 100, or -7) is always zero because flat numbers do not change.

Integration is Detective Work

Integration is the exact reverse of a derivative. It asks: “What original equation gave me $2x$?’

When you look at 2x, you know the original equation started with x². However, you have no way of knowing if there used to be a hidden number attached to it (like the +5 or -12) because the derivative destroyed it.

What +C Actually Means

Since we cannot guess what the original mystery number was, we write +C to say: “There was likely a constant number here, but it became zero, so I am using $C$ as a placeholder.”