What is the integral of x^(3)e^(x) with respect to x?

This question is looking at integration by parts. Therefore, we need to split the integral into two parts.

Part A will be x³ and Part B will be e^x .

Draw 2 columns putting Part A in the RIGHT and B in the LEFT. (Positions will matter later).

Differentiate part A until it reaches 0 so dy/dx of x³=3x² then dy/dx of 3x²=6x and dy/dx of 6x=6 and dy/dx of 6=0.

Then, integrate e^x by the same amount of times as we differentiated x³ . The integral of e^x=e^x .

Here we should have 5 rows and 2 columns.

Now, draw diagonal lines in a NE direction from the position row 2, left column to row 1, right column. This should match  e^x (on the second row) with 3x² . Repeat this step drawing NE facing lines until you reach the end of the table. This should leave 2 'unmatched' items which are 0 (bottom of right column) and e^x (top of left column).

Now write alternating + and - (starting with + and then - and then + and then -) along the lines and multiply along the lines and don't forget to +c at the end.

This should leave us with x³e^x - 3x²e^x + 6xe^x - 6e^x + c

This can be simplified to e^x(x³ - 3x² + 6x - 6) + c