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Discussion Forum » MSC »

Integral of X^4/((X-1) * (X-2))

I'm doing this for homework and I can't figure out how to simplify X^4/((X-1) * (X-2)). Please help.

-- Main.jts228 - 2009-10-07

(x-1)*(x-2) = x^2-3x+2. So, we can do polynomial long division to get:

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Then, we can do partial fraction decomposition on to find an equivalent expression of the form Clearing denominators gives us the equation 15x-14=A(x-2)+B(x-1), which we can rearrange as 15x-14=(A+B)x-(2A+B). Then, equate the x coefficients on both sides, and the constant expressions on both sides to get the system of equations A+B=15, 14=2A+B. We can solve these to get A=-1, B=16. So,
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Then, we can do partial fraction decomposition on
to find an equivalent expression of the form
Clearing denominators gives us the equation
15x-14=A(x-2)+B(x-1), which we can rearrange as
=(A+B)x-(2A+B).
Then, equate the x coefficients on both sides, and the constant expressions on both sides to get the system of equations A+B=15,
2A+B=14.
We can solve these to get A=-1, B=16. So,
 
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Combining all this, we get
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Combining all this, we get
.
Which can be integrated term-by-term.
  -- MattGuay - 2009-10-07
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See also:

...and for the details of the partial fractions decomposition:

partialfractions.png

in the initial box with the short answer, each of these links invites you to Show steps which reveals possible steps to arrive at the solution. The entire partial fractions decomposition in the first link is implicit in the long division of its first step.

-- DickFurnas - 2010-01-26

 
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