Tabular Integration (The Tabular Method)

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What is Tabular Integration?

Tabular integration is a different way to tackle integration by parts problems. While it’s more straightforward than using the integration by parts formula, it doesn’t work for all problems. In order for this method to work, the term you pick for “u” has to eventually become zero when you take successive derivatives.

Tabular Integration Example

Example question: Solve ∫(x³ + 2x – 1) cos(4x) with tabular integration.

Step 1: Create a two column table. Label the first column u and the second column dv (these is standard integration by parts notation:

Step 2: In the first row, place your choices for u and v. There are two parts to this function: (x³ + 2x – 1) and cos(4x). It’s not always clear which is the best choice for u. If you aren’t sure which one to pick, just try one. You can always redo the table with your second choice.

Step 3: In the first column, take the derivative (not the antiderivative!)

n

times until you reach zero. For this example, we had to go up to the fourth derivative.


If you don’t get to zero (for example, if your derivatives start to repeat or if there’s no end in sight), then switch the columns around and try differentiating the other part of the function:

Step 4: Fill in the derivatives for the second column in the same way. Stop when you reach the same number of rows as the first column (in other words, stop at 0):

Step 5: Place arrows down and to the right from each entry in the first column to the next lowest entry in the second column. These indicate where we will be multiplying:

Step 6: Add alternating “+” and “-” signs to the arrows, starting with “+”:

Step 7: Multiply the terms in the table, according to the arrows and the ± signs:

Step 8: List all of your answers:

Step 9: Tidy up the solution and add a C:


That’s it!

References

Calculus 2: Integration Techniques.
Daileda, R. (2018) The Tabular Method for Repeated Integration by Parts. Retrieved July 11, 2021 from: http://ramanujan.math.trinity.edu/rdaileda/teach/s18/m3357/parts.pdf
Rock, J, It’s just parts: A user’s guide for the tabular method of integration by parts. Retrieved July 11, 2021 from: https://www.cpp.edu/~jarock/RIP_JMM_180113.pdf