Tabular integration

A shortcut for integration by parts applied repeatedly, via a two-column derivative/integral table.

Tabular integration streamlines REPEATED integration by parts (needed when $u$ must be differentiated more than once, e.g. a polynomial factor) into a two-column table: one column differentiates $u$ down to zero, the other integrates $dv$ the same number of times, and the answer is the alternating-sign sum of diagonal products.

Examples

For $\int x^2 e^x\,dx$ , the table differentiates $x^2\to2x\to2\to0$ while integrating $e^x\to e^x\to e^x$ , giving $(x^2-2x+2)e^x+C$ .