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Difficult Calculus II integral

  1. Oct 13, 2011 #1
    1. The problem statement, all variables and given/known data
    [tex]\int_{-1}^{0} \frac{e^{\frac{1}{x}}}{x^{3}}dx[/tex]
    Solve the integral and determine if it converges/diverges ect.

    2. Relevant equations

    3. The attempt at a solution
    [tex]\int_{-1}^{0} \frac{e^{\frac{1}{x}}}{x^{3}}dx[/tex]
    [tex]\lim_{t \to 0}\int_{-1}^{t} \frac{e^{\frac{1}{x}}}{x^{3}}dx[/tex]

    [itex]u = \frac{1}{x}[/itex]
    [itex]du = \frac{1}{x^{2}}dx[/itex]

    [tex]\lim_{t \to 0}\int_{-1}^{\frac{1}{t}} \frac{e^{u}(-x^{2})}{x^{3}}du[/tex]
    [tex]\lim_{t \to 0}\int_{-1}^{\frac{1}{t}} \frac{-e^{u}}{x} du[/tex]
    [tex]x = \frac{1}{u}[/tex]
    [tex]\lim_{t \to 0}\int_{-1}^{\frac{1}{t}} \frac{-e^{u}}{\frac{1}{u}} du[/tex]
    [tex]\lim_{t \to 0} \int_{-1}^{\frac{1}{t}} -e^{u}u du[/tex]

    [itex]v = u[/itex]
    [itex]dv = du[/itex]
    [itex]w = -e^{u}[/itex]
    [itex]dw = -e^{u}du[/itex]

    [tex]\lim_{t \to 0} (-e^{u}u)_{-1}^{\frac{1}{t}} - \int_{-1}^{\frac{1}{t}} -e^{u}du[/tex]

    [tex]\lim_{t \to 0} (\frac{-e^{\frac{1}{t}}}{t} + e^{-1}(-1)) - (-e^{\frac{1}{t}}+e^{-1})[/tex]

    [tex]\lim_{t \to 0} (\frac{-e^{\frac{1}{t}}}{t}-e^{-1}) + e^{\frac{1}{t}}-e^{-1}[/tex]

    [tex]\lim_{t \to 0} (e^{\frac{1}{t}}-\frac{e^{\frac{1}{t}}}{t}-2e^{-1})[/tex]

    And now I can't solve this limit. What am I doing wrong here? I put the limit into my TI-89 and as I thought it is undefined. I know the solution is that it converges to [itex]\frac{-2}{e}[/itex] but I can't seem to get there.
     
    Last edited: Oct 13, 2011
  2. jcsd
  3. Oct 13, 2011 #2

    SammyS

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    The indefinite integral, [itex]\displaystyle \int\, \frac{e^{\frac{1}{x}}}{x^{3}}dx[/itex]

    becomes [itex]\displaystyle -e^{u}u - \int\, -e^{u}du\,,[/itex] where u = 1/x

    [itex]\displaystyle = -e^{u}u + e^u
    +C\,.[/itex]

    Back substituting (u=1/x) gives:

    [itex]\displaystyle \int\, \frac{e^{\frac{1}{x}}}{x^{3}}dx=-\frac{e^{1/x}}{x}+e^{1/x}+C[/itex]

    Now, evaluate [itex]\displaystyle \lim_{t \to 0}\int_{-1}^{t} \frac{e^{\frac{1}{x}}}{x^{3}}dx \,.[/itex]
     
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