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Thread: Problem with 2nd degree polynomial with the integral being a rational function

  1. #1
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    Problem with 2nd degree polynomial with the integral being a rational function

    The problem goes like this:

    Find the second degree polynomial P(x) such that P(0) = 1, P'(0) = 0, and the integral of P(x) over x^3(x-1)^2 is a rational function


    I got stuck after I solved for c and b in the ax^2+bx+c with b=0 and c=1

    there are hints attached Problem with 2nd degree polynomial with the integral being a rational function-screen-shot-2017-03-08-12.08.19-pm.png
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  2. #2
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    Re: Problem with 2nd degree polynomial with the integral being a rational function

    Your attachment doesn't have anything to do with this question! Yes, a quadratic polynomial, P(x), such that P(0)= 1 and P'(0)= 0, is of the form P(x)= ax^2+ 1. The third condition is that "the integral of P(x) over x^3(x-1)^2 is a rational function".

    That is \int \frac{ax^2+ 1}{x^3(x- 1)^2}dx. Do that integration by "partial fractions". We can write \frac{ax^2+ 1}{x^3(x- 1)^2} as \frac{A}{x}+ \frac{B}{x^2}+ \frac{C}{x^3}+ \frac{D}{x- 1}+ \frac{E}{(x- 1)^2}.

    The crucial point is that the integrals of \frac{A}{x} and \frac{D}{x- 1} are logarithms, not rational functions. Can you find "a" such that A and D are 0?
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