How to Solve Problems of Integration by Parts Using the Tabular Method

Doing calculus problems which require integration by parts can be a lengthy and tedious process, even for someone with experience in finding the integrals of functions. Fortunately, for many of the most common types of integration by parts, there is fast, simple shortcut available. I have used the tabular method to great advantage on exams and math contests; this technique was one of the tools that enabled me to earn a perfect score of 40 on the Continental Mathematics League's nationwide calculus competition in 2005.

The tabular method can be applied to any function which is the product of two expressions, where one of the expressions has some nth derivative equal to zero. For instance, the tabular method can be used to find the indefinite integral of x⁴e^3x, but not of sin(x)e^3x. This is because the 5^th derivative of x⁴ is equal to zero, whereas sin(x) does not have any nth derivative which always exhibits zero values.

The tabular method uses a convenient table with three columns. We can call the first column "Signs," the second column "u" and the third column "dv." Under the column called "Signs," we list positive and negative signs in alternating order for as many times as the problem requires. The first entry in the column labeled "u" will be the part of the function we want to integrate which can be reduced to zero through successive differentiation. In integrating our sample function, x⁴e^3x, we will put x⁴ in the column labeled "u." The subsequent entries in the "u" column will be the successive derivatives of the first entry -- all the way to zero.

The first entry in the column labeled "dv" includes the other part of the function we want to integrate. The subsequent entries in this column will be the successive integrals of the first entry. For our sample problem, the first entry in the "dv" column will be e^3x.

This is how the table for finding the integral of x⁴e^3x will look:

Signs....... u.................. dv
+............... x⁴................ e^3x
- ...............4x³ .......(1/3)e^3x
+.............12x² .......(1/9)e^3x
- ............. 24x ........(1/27)e^3x
+.............24............(1/81)e^3x
-............... 0...........(1/243)e^3x

Using this table to find the indefinite integral of the function requires taking the sign from each row in the column except the last, applying it to the entry for "u" in the same row, and multiplying the result by the entry for "dv" in the next row. Doing this problem on paper, you would simply draw arrows between the following expressions:

+ x⁴ and (1/3)e^3x
- 4x³ and (1/9)e^3x
+ 12x² and (1/27)e^3x
- 24x and (1/81)e^3x
+ 24 and (1/243)e^3x

Now multiply each of the expressions linked by "and" (or the arrows on paper) and add them together to get the final indefinite integral:

(1/3)x⁴e^3x - (4/9)x³e^3x + (4/9)x²e^3x - (8/27)xe^3x + (8/81)e^3x + C

Remember to include the constant C if you wish to leave the integral in indefinite form without evaluating it.

The great advantage of using the tabular method is the ability to do integration by parts mechanically, without needing to exert a large amount of thinking about the special circumstances of the problem. The procedure is fairly quick to memorize and easy to retain. After you learn it once, it will always be at your disposal as a tool for quickly and easily determining indefinite integrals that would otherwise take an immense amount of time to find.