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Thread: Help with integration with partial fractions

  1. #1
    Junior Member TriForce's Avatar
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    Help with integration with partial fractions

    So i need to integrate:

    $ \frac{5x^3+12x^2+12x+10}{(x^2+4)(x^2+2x+1)}$

    $ =\frac{5x^3+12x^2+12x+10}{(x^2+4)(x+1)^2}$

    Partial fractions attempt:

    $ \frac{Ax+B}{x^2+4}$ + $\frac{C}{x+1}$ + $\frac{D}{(x+1)^2}$

    $(Ax+B)(x+1)^2=(Ax+B)(x^2+2x+1)=Ax^3+2Ax^2+Ax+Bx^2 +2Bx+B$
    $C(x^2+4)(x+1)=C(x^3+x^2+4x+4)=Cx^3+Cx^2+4Cx+4C$
    $D(x^2+4)=Dx^2+4D$

    $x^3(A+C)=5$

    $x^2(2A+B+C+D)=12$

    $x^1(A+2B+4C+4D)=12$

    $x^0(B+4C+4D)=10$

    $A=\frac{32}{5}$
    $B=-\frac{22}{5}$
    $C=-\frac{7}{5}$
    $D=5$

    Is this wrong so far? Mathematica seems to think so.
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  2. #2
    MHF Contributor
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    Re: Help with integration with partial fractions

    Try it out.

    Wolfram|Alpha: Computational Knowledge Engine)

    It is not identically zero, so you made a mistake somewhere.

    Let's go through it a bit more carefully.

    $\begin{matrix} & Ax^3 & + & (2A+B)x^2 & + & (A+2B)x & + & B\\+ & Cx^3 & + & Cx^2 & + & 4Cx & + & 4C \\ & & + & Dx^2 & & & + & 4D \\ \hline & (A+C)x^3 & + & (2A+B+C+D)x^2 & + & (A+2B+4C)x & + & (B+4C+4D)\end{matrix}$

    The difference between this chart and what you wrote is with the $x^1$-term.
    Last edited by SlipEternal; Feb 20th 2018 at 07:32 AM.
    Thanks from TriForce
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  3. #3
    MHF Contributor

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    Re: Help with integration with partial fractions

    Quote Originally Posted by TriForce View Post
    So i need to integrate:

    $ \frac{5x^3+12x^2+12x+10}{(x^2+4)(x^2+2x+1)}$

    $ =\frac{5x^3+12x^2+12x+10}{(x^2+4)(x+1)^2}$

    Partial fractions attempt:

    $ \frac{Ax+B}{x^2+4}$ + $\frac{C}{x+1}$ + $\frac{D}{(x+1)^2}$

    $(Ax+B)(x+1)^2=(Ax+B)(x^2+2x+1)=Ax^3+2Ax^2+Ax+Bx^2 +2Bx+B$
    $C(x^2+4)(x+1)=C(x^3+x^2+4x+4)=Cx^3+Cx^2+4Cx+4C$
    $D(x^2+4)=Dx^2+4D$

    $x^3(A+C)=5$

    $x^2(2A+B+C+D)=12$

    $x^1(A+2B+4C+4D)=12$

    $x^0(B+4C+4D)=10$

    $A=\frac{32}{5}$
    $B=-\frac{22}{5}$
    $C=-\frac{7}{5}$
    $D=5$

    Is this wrong so far? Mathematica seems to think so.
    I hate this kind of busywork. I just use what I can. SEE HERE
    Thanks from TriForce
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  4. #4
    Junior Member TriForce's Avatar
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    Re: Help with integration with partial fractions

    @SlipEternal
    This is very helpful. I'll give it another try!
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