Expert: Paul Klarreich - 6/16/2006

Question
I am trying to simplify something so that I can use a table of integrals to integrate.

The expression is:

(x^3 + x^2 - x + 1) / (x^2 + 9) dx

I think I'm supposed to make it look like du/(u^2 + a^2) ... and similar problems in the book show them doing "long division" to simplify these types of expressions (I guess I'm trying to eliminate x in the numerator)...but I'm not sure if that's even the simplification I'm working for. There's another one where the denominator is to the power of 3/2 - maybe that's where I'm going?

Anyway, I'm hitting a brick wall with trying to simplify this. Here I am doing calculus and I'm forgetting my 8th grade algebra.

Answer
Hi, Abby,

You wrote:
Subject:  algebra - long division?, also integration using tables
Question:  I am trying to simplify something so that I can use a table of integrals to integrate.

The expression is:

(x^3 + x^2 - x + 1) / (x^2 + 9) dx

I think I'm supposed to make it look like du/(u^2 + a^2) ... and similar problems in the book show them doing "long division" to simplify these types of expressions

(I guess I'm trying to eliminate x in the numerator)...but I'm not sure if that's even the simplification I'm working for. There's another one where the denominator is to the power of 3/2 - maybe that's where I'm going?

Anyway, I'm hitting a brick wall with trying to simplify this. Here I am doing calculus and I'm forgetting my 8th grade algebra.
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I think you have written me with questions before.  In that case, you should know by now that you have to:

USE A FIXED SIZE FONT TO VIEW THIS, BECAUSE THERE ARE FRACTIONS.

First of all, this is 9th grade algebra, not 8th.  Second, you are correct in that:

1. You should do long division.
2. You will have a  du/(u^2 + a^2) type integral.

However, you should expect THREE pieces to appear and be integrated:

A. A quotient that is just a polynomial. (easy)
B. A remainder term that has 'x'.  It will look like

{    Ax
| -------- dx
} x^2 + 9

and you will integrate by letting u = x^2 + 9, and get a log expression.

C. A remainder term that is a constant.  It will look like:

{    B
| -------- dx
} x^2 + 9

and this will be as you described, and produce a result with arctangent in it.

So it's just a matter of bringing over the ladder and climbing the wall instead of trying to go through it.  Divide: (this is how MY 9th grade teacher said to do it)

x^2 + 9  )  x^3 + x^2 - x + 1  ( x + 1
           x^3       +9x
          -------------------
                 x^2 -10x + 1
                 x^2      + 9
                 -------------
                     -10x - 8

So:
                           -10x - 8
x^3 + x^2 - x + 1 = x + 1 + ---------
                            x^2 + 9  

Now do three integrals.  I'll leave this to you.  It's messy and not so much fun, but it's routine.

{
| (x + 1) dx   ...... the easy one.
}

{ -10x dx
| --------  ........ should have  ln(x^2 + 9) in the answer
} x^2 + 9

}  - 8 dx
| --------  ........ will have arctangent in the answer.
} x^2 + 9