AP® Calculus BC · Unit 6 · Topic 6.11
Repeated Integration by Parts and the Tabular Method
Repeated integration by parts lets you peel off a polynomial layer by layer, or catch a repeating integral and solve for it algebraically; the tabular method just organizes multiple rounds of parts into one fast table.
Integration by parts turns the integral of u dv into uv minus the integral of v du. Sometimes that new integral is still a product you can't do directly, so you apply parts again. There are two ways this plays out. If one factor is a polynomial like x2, each round of parts lowers its power by one, until it vanishes entirely and leaves a plain integral. If instead the factors are things like ex and cos(x), differentiating and integrating never simplifies anything - after two rounds the original integral reappears on the right side. That's not failure. It means you now have an equation with the same integral on both sides, and you can solve for it with algebra, just like solving for x.
The tabular method is a shortcut for the polynomial case. Make two columns: one where you repeatedly differentiate the polynomial factor down to zero, another where you repeatedly integrate the other factor. Multiply diagonally, alternating signs (+, -, +, -...), and add up the diagonal products. This replaces writing out three separate by-parts steps with one table, and it's especially fast when the polynomial has degree three or four.
This topic is the capstone of the by-parts techniques in the unit: it shows that the method is flexible enough to handle both terminating products and self-referential ones. It also previews a habit used later with differential equations and series, where you often manipulate an equation containing the unknown quantity on both sides to isolate it, rather than evaluating directly.
Key terms
- Integration by parts
- A technique rewriting integral of u dv as uv minus integral of v du, based on the product rule.
- Tabular method
- A table-based shortcut for repeated integration by parts using alternating signs along diagonal products.
- Repeated integral
- A case where applying integration by parts twice returns the original integral, allowing it to be solved for algebraically.
- Antiderivative
- A function whose derivative equals the original integrand, determined up to a constant.
Watch it explained
- How it worksHow does a tabular layout of derivatives and antiderivatives shortcut three rounds of parts?
- Worked exampleFind the integral of x2 sin(x) dx by using parts twice.
- Worked exampleFind the integral of ex cos(x) dx by solving for the repeated integral.
- Common mistakeWhy isn't a repeating integral a sign that integration by parts has failed?
Check yourself
End of topic.