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Integral Problem

  1. Mar 1, 2010 #1
    1. The problem statement, all variables and given/known data

    [tex]\int\frac{xdx}{3+\sqrt{x}}[/tex]


    2. Relevant equations

    The answer is given: [tex]\frac{2}{3}x^\frac{3}{2}-3x+18\sqrt{x}-54ln(3+\sqrt{x})+C[/tex]

    3. The attempt at a solution

    [tex]u=\sqrt{x}[/tex]

    [tex]u^2=x[/tex]

    [tex]2udu=dx[/tex]

    [tex]\int\frac{xdx}{3+\sqrt{x}} = 2\int\frac{(u^3)du}{3+u}[/tex]

    [tex]w=3+u[/tex]

    [tex]w-3=u[/tex]

    [tex]dw=du[/tex]

    [tex]=2\int\frac{(w-3)^3dw}{w}[/tex]

    [tex]=2\int\frac{(w^3-9w^2+27w-27)dw}{w}[/tex]

    [tex]=2\int\((w^2-9w+27-\frac{27}{w})dw[/tex]

    [tex]=2\int\(w^2dw-18\int\(wdw+54\int\(dw-54\int\frac{dw}{w}[/tex]

    [tex]=2\frac{w^3}{3}-18\frac{w^2}{2}+54w-54ln|w|+C[/tex]

    [tex]=\frac{2}{3}(3+u)^3-9(3+u)^2+54(3+u)-54ln|3+u|+C[/tex]


    [tex]=\frac{2}{3}(3+\sqrt{x})^3-9(3+\sqrt{x})^2+54(3+\sqrt{x})-54ln|3+\sqrt{x}|+C[/tex]

    I multiplied this out but terms didn't cancel. Any suggestions?
     
  2. jcsd
  3. Mar 1, 2010 #2
    looks like you got everything right, Im sure you made some calculation mistake.

    what's your final answer ?
     
  4. Mar 1, 2010 #3
    If you multiply out those terms, you'll get the given answer, but with an additional term of + 99. (or something of that sort)

    Assuming this is your problem, you just need to remember that the constant of integration C is arbitrary, so it can "absorb" any constant terms.
     
  5. Mar 1, 2010 #4
    Alright it looks like I just made a mistake last time:

    [tex]=\frac{2}{3}(3+\sqrt{x})^3-9(3+\sqrt{x})^2+54(3+\sqrt{x})-54ln|3+\sqrt{x}|+C[/tex]

    [tex]=\frac{2}{3}(3+\sqrt{x})(3+\sqrt{x})(3+\sqrt{x})-9(3+\sqrt{x})(3+\sqrt{x})+54(3+\sqrt{x})-54ln|3+\sqrt{x}|+C[/tex]

    [tex]=\frac{2}{3}(9+6\sqrt{x}+x)(3+\sqrt{x})-9(9+6\sqrt{x}+x)+54(3+\sqrt{x})-54ln|3+\sqrt{x}|+C[/tex]

    [tex]=\frac{2}{3}(27+9\sqrt{x}+18\sqrt{x}+6x+3x+x^\frac{3}{2})-81-54\sqrt{x}-9x+162+54\sqrt{x}-54ln|3+\sqrt{x}|+C[/tex]

    [tex]=\frac{2}{3}(27+9\sqrt{x}+18\sqrt{x}+6x+3x+x^\frac{3}{2})-81+162-54\sqrt{x}+54\sqrt{x}-9x-54ln|3+\sqrt{x}|+C[/tex]


    [tex]=\frac{2}{3}(27+27\sqrt{x}+9x+x^\frac{3}{2})+81-9x-54ln|3+\sqrt{x}|+C[/tex]

    [tex]=18+18\sqrt{x}+6x+\frac{2}{3}x^\frac{3}{2}+81-9x-54ln|3+\sqrt{x}|+C[/tex]

    [tex]=99+18\sqrt{x}-3x+\frac{2}{3}x^\frac{3}{2}-54ln|3+\sqrt{x}|+C[/tex]

    [tex]=\frac{2}{3}x^\frac{3}{2}-3x+18\sqrt{x}-54ln|3+\sqrt{x}|+C[/tex]

    Thank you for the help!
     
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