Answer to Exercise 11.4-4

Step 1: Choosing your parts is done for you in the statement of the problem. It is

b)
u = x
and
dv = x e^-x dx

Step 2: What is v? Just integrate dv/dt, which is done for you in the statement of the problem as well (although you would have needed another round of integration by parts at this step to determine v if this integral weren't already given).

v =

x e^-x dx =

-x e^-x - e^-x + C

Plug what you have for u and v into equation 11.4-6a, then click here to see what you get.

You should have

-x² e^-x - x e^-x + C -

-x e^-x - e^-x

du dx = dx

x² e^-x dx

or perhaps you cancelled a few minus signs and got

-x² e^-x - x e^-x + C +

x e^-x + e^-x

du dx = dx

x² e^-x dx

Step 3: What is du/dx? Just take the derivative of u = x. And since you've seen this one a bunch of times already, I'm not going to make you click for it.

   du
       =  1
   dx

which simplifies things a bit. Now you have

-x² e^-x - x e^-x + C +

(x e^-x + e^-x) dx =

x² e^-x dx

Step 4: Do the remaining integral. It is an integral of a sum, so you can apply equation 11.2-13b to it. That turns it into

-x² e^-x - x e^-x + C +

x e^-x dx +

e^-x dx =

x² e^-x dx

You already can figure the second integral from the table.

e^-x dx = -e^x + C

and the first one was given in the problem

x e^-x dx =

-x e^-x - e^-x + C

So just substitute those expressions in, gather like terms, and then check to make sure you got the same thing as we got when we did it in the text.

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