Method for integrating by parts -- if the differential of one of the functions eventually equals zero.
Construct three columns -- one for the function you will differentiate (u), one for the function you will integrate (dv), and one for alternating signs + and -. Differentiate the function until it equals zero. Your answer is found by summing the products of three quantities in each diagonal (from top left to bottom right) that has three entries in it.
Recall the formula for integrating by parts:
Example #1:
Using Escalante's method:
| Differentiate: | Integrate: | Signs: |
| x2 | sin(x) | + |
| 2x | -cos(x) | - |
| 2 | -sin(x) | + |
| 0 | cos(x) | - |
| + |
You get this same answer by taking the taking the products of the three quantities in each of the diagonals (shown in color) and then adding these three products together.
Example #2:
Using Escalante's method:
| Differentiate: | Integrate: | Signs: |
| x2 | ex | + |
| 2x | ex | - |
| 2 | ex | + |
| 0 | ex | - |
| + |