Repeated and tabular integration by parts

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

If repeated polynomial differentiation reaches zero, organize derivatives and primitives of the other factor in a table with alternating signs. The table summarizes repeated applications of the same formula.

Step-by-step example

1. Differentiate the polynomial to zero.

   $$x^2,\ 2x,\ 2,\ 0$$

2. Integrate eˣ repeatedly and alternate signs.

   $$e^x(x^2)-e^x(2x)+e^x(2)$$

3. Add C and check by differentiating.

   $$\int x^2e^x\,dx=e^x(x^2-2x+2)+C$$