Use polynomial long division to rewrite and integrate...

[Jaime_Kartel]

This is the problem:

Use polynomial long division to rewrite and integrate ∫(x^2) / (x-2) dx

I thought it would be as simple as doing normal long division
_x_+_2x/(x-2)_
x-2 | x^2
-x^2 - 2x
0 + 2x


But it turns out it is not, and I cannot figure out why. I have the answer to the question, but I do not know how it is arrived.

 

[Deleted member 4993]

This is the problem:

Use polynomial long division to rewrite and integrate ∫(x^2) / (x-2) dx

I thought it would be as simple as doing normal long division
_x_+_2x/(x-2)_
x-2 | x^2
-x^2 - 2x
0 + 2x


But it turns out it is not, and I cannot figure out why. I have the answer to the question, but I do not know how it is arrived.

But it is!!

What did you get as the result (I cannot decipher your work)

 

[stapel]

Use polynomial long division to rewrite and integrate ∫(x^2) / (x-2) dx

I thought it would be as simple as doing normal long division
_x_+_2x/(x-2)_
x-2 | x^2
-x^2 - 2x
0 + 2x

But it turns out it is not, and I cannot figure out why.

I think you mean your long division to look like this:

long division:

     x + 2
    -------------
x-2 )x^2 + 0x + 0
     x^2 - 2x
     ------------
           2x + 0
           2x - 4
           ------
                4

Either way, what do you mean, specifically, when you say that "doing normal [polynomial] long division" is not the correct way to do the stipulated "polynomial long division"?

By the way, I can't understand your "work" either. My steps were as follows:

set up the division:

    -------------
x-2 )x^2 + 0x + 0


to get x^2, multiply x by x (and put it on top):

     x 
    -------------
x-2 )x^2 + 0x + 0


carry this down:

     x 
    -------------
x-2 )x^2 + 0x + 0
     x^2 - 2x


subtract the new line from the original line:

     x 
    -------------
x-2 )x^2 + 0x + 0
     x^2 - 2x
     ------------
           2x + 0


to get 2x, multiply x by 2 (and put it on top):

     x + 2
    -------------
x-2 )x^2 + 0x + 0
     x^2 - 2x
     ------------
           2x + 0


carry this down:

     x + 2
    -------------
x-2 )x^2 + 0x + 0
     x^2 - 2x
     ------------
           2x + 0
           2x - 4


subtract the new line from the previous line:

     x + 2
    -------------
x-2 )x^2 + 0x + 0
     x^2 - 2x
     ------------
           2x + 0
           2x - 4
           ------
                4


so the integrand converts as:

 x^2              4
----- = x + 2 + -----
x - 2           x - 2

 

[Ishuda]

This is the problem:

Use polynomial long division to rewrite and integrate ∫(x^2) / (x-2) dx

I thought it would be as simple as doing normal long division
_x_+_2x/(x-2)_
x-2 | x^2
-x^2 - 2x
0 + 2x


But it turns out it is not, and I cannot figure out why. I have the answer to the question, but I do not know how it is arrived.

Keep going.
HINT: $\displaystyle \dfrac{x^2}{x-2}\, =\, \dfrac{x^2 - 4 + 4}{x-2}\, =\, \dfrac{x^2 - 4}{x-2}\, +\, \dfrac{4}{x-2}$