1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    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]
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook