The integral of x^2 e^x

To use parts to compute the integral of x^2 e^x, I have to divide the stuff in the integral up into two chunks. The product of the two chunks must be x^2 e^x, but otherwise I'm free to do the division as I please. (Of course, some divisions will work better than others.)

One chunk corresponds to "u" in the parts formula; the other chunk corresponds to "dv". I'll let u = x^2 and dv = e^x dx. However, you drop the "dx" and just write the "e^x" in the table.

My table has three columns. The derivative column (the one with the "d/dx") starts with x^2 (my "u") and the integral column starts with e^x (my "dv", without the "dx"). Watch the animation below: It shows how I'd fill in the table, step by step.

1. I put x^2 ("u") into the derivative column and e^x ("dv", without the "dx") into the integral column.
2. I filled in the first column with alternating plus and minus signs.
3. Next, I differentiated x^2 to get 2x, differentiated 2x to get 2, and differentiated 2 to get 0. (When I get 0, I can stop.) You differentiate the stuff in the derivative column.
4. After that, I integrated e^x to get e^x, integrated e^x to get e^x, and integrated e^x to get e^x. You integrate the stuff in the integral column.
5. Finally, I used the parts pattern to get the successive terms in the answer. I was careful to remember the "+ C", since I'm computing an antiderivative (an improper integral).

What about the "0" in the derivative column? Why does this mean that I can stop my table after 4 rows?

Think back to the discussion of where the parts table comes from. Recall that you can stop a table at any point by integrating the product of the terms in the last row. In this case, the product of the terms in the last row is 0, and the antiderivative of 0 is a constant. But constants are accounted for by the arbitrary constant --- the "+ C".

Thus, if you get 0 by differentiating, your table stops automatically at the 0-row. This is usually a good thing.

Finally, why would you think of using integration by parts to do this integral? Notice that the integral contained x^2 and e^x, a power and an exponential. Parts is often useful when an integral contains different kinds of functions.

• Introduction: Integration by parts
• How do you set up an integration by parts table?
• The integral of x^2 e^x
  • How do you choose what to put in the table?
• The integral of ln x
• The integral of e^(2x) cos (3x)
• A recursion formula for the integral of (sin x)^n
• Computing a definite integral using integration by parts

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Last updated: June 13, 2005

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