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Partial fractions

  1. Oct 11, 2009 #1
    [tex]
    \int \frac{5x^2+11x+17}{x^3+5x^2+4x+20}dx
    [/tex]
    [tex]
    \int \frac{5x^2+11x+17}{(x^2+4)(x+5)}dx
    [/tex]
    [tex]
    \frac{Ax+B}{x^2+4}+\frac{C}{x+5}=\frac{5x^2+11x+17}{(x^2+4)(x+5)}
    [/tex]
    [tex]
    (Ax+B)(x+5)+C(x^2+4)=5x^2+11x+17
    [/tex]
    [tex]
    Ax^2+5Ax+Bx+5B+Cx^2+4C=5x^2+11x+17
    [/tex]
    [tex]
    x^2(A+C)+x(5A+B)+(5B+4C)=5x^2+11x+17
    [/tex]
    [tex]
    A+C=5, 5A+B=11, 5B+4C=17
    [/tex]
    [tex]
    A=5-C
    [/tex]
    [tex]
    5(5-C)+B=11, 25-5C+B=11, B=-14+5C
    [/tex]
    [tex]
    5(-14+5C)+4C=17, -70+29C=17, C=3, B=1, A=2
    [/tex]
    [tex]
    \int \frac{2x+1}{x^2+4}+ \frac{3}{x+5}dx
    [/tex]
    [tex]
    ln(x^2+4) +aractan(x/2)/2+3ln|x+5|+Z
    [/tex]
    orginally I thought I had made a mistake somwhere but I beleive this is correct please make suggestions im new to this technique
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 11, 2009 #2

    Mark44

    Staff: Mentor

    Looks good, but I haven't checked each detail. Two things you can do are
    1) check that (2x + 1)/(x2 + 4) + 3/(x + 5) = your original integrand.
    2) check that d/dx[ln(x2 + 4) + 1/2*arctan(x/2) + ln|x + 5| = your original integrand.
     
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