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 ex .

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 ex by the same amount of times as we differentiated x3 . The integral of ex=ex .

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  ex (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 ex (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- 3x2ex + 6xex - 6ex + c

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

Answered by James B. Maths tutor

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