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: