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Math Help - integration via partial fractions

  1. #1
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    integration via partial fractions

    \int \frac{x^3 +x - 2}{(x^2+1)(x^2+2)} dx
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  2. #2
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    x^3+x-2=(Ax+B)(x^2+2)+(Cx+D)(x^2+1)
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  3. #3
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    i had that but how do u solve for each of the 4 variables?
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  4. #4
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    x^3+x-2=(Ax+B)(x^2+2)+(Cx+D)(x^2+1)



    expand so


    x^3+x-2=Ax^3+2Ax+Bx^2+2B+Cx^3+Cx+Dx^2+D

    so 1x^3+x-2=(A+C)x^3+(B+D)x^2+(2A+C)x+(2B+D)


    so (set them equal to the coefficients)

    A+C=1
    B+D=0
    2A+C=1
    2B+D=-2


    you have two equations for 2 variables in both cases let A=1-C for example and substitue in so 2(1-C)+C=1 so 2-C=1 C=1 so A=0


    i'm doing this on the computer i could be wrong though..my math that is
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  5. #5
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    i got the answer to be:

     -2 tan^{-1} (x) + \frac{1}{2} ln | x^2 + 2 | + \frac{2}{\sqrt{2}} tan^{-1} \left( \frac{x}{\sqrt{2}}\right) + C
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  6. #6
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    that is correct
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