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Math Help - integrate integral 1/x(ln (x)^1/2)

  1. #1
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    Question integrate integral 1/x(ln (x)^1/2)

    Question asks to evaluate the integral
    <br />
\int\limits_e^{e^4 } {\frac{1}{{x\sqrt {\ln x} }}} dx<br />

    Attempt by substitution, Letting u = ln x
    <br />
 = \int\limits_e^{e^4 } {\frac{1}{x}u^{\frac{{ - 1}}{2}} dx} <br />

    and so
    <br />
\begin{array}{l}<br />
 \frac{{du}}{{dx}} = \frac{1}{x} \\ <br />
 du = \frac{1}{x}dx \\ <br />
 \end{array}<br />

    giving integral as

    <br />
 = \int\limits_e^{e^4 } {u^{\frac{{ - 1}}{2}} du} <br />

    when I integrate this last integral I get this

    <br />
\begin{array}{l}<br />
  = \left[ {\frac{{u^{\frac{1}{2}} }}{{\frac{1}{2}}}} \right]_e^{e^4 }  \\ <br />
  = \left[ {2u^{\frac{1}{2}} } \right]_e^{e^4 }  \\ <br />
  = \left[ {2(\ln x)^{\frac{1}{2}} } \right]_e^{e^4 }  \\ <br />
  = 2(\ln e^4 )^{\frac{1}{2}}  - 2(\ln e)^{\frac{1}{2}}  \\ <br />
  = 4 - 1 \\ <br />
  = 3 \\ <br />
 \end{array}<br />

    The answer is given as 2, I'm not sure what I have done wrong here ?
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  2. #2
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    earboth's Avatar
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    Quote Originally Posted by Craka View Post
    Question asks to evaluate the integral
    <br />
\int\limits_e^{e^4 } {\frac{1}{{x\sqrt {\ln x} }}} dx<br />

    Attempt by substitution, Letting u = ln x
    <br />
 = \int\limits_e^{e^4 } {\frac{1}{x}u^{\frac{{ - 1}}{2}} dx} <br />

    and so
    <br />
\begin{array}{l}<br />
 \frac{{du}}{{dx}} = \frac{1}{x} \\ <br />
 du = \frac{1}{x}dx \\ <br />
 \end{array}<br />

    giving integral as

    <br />
 = \int\limits_e^{e^4 } {u^{\frac{{ - 1}}{2}} du} <br />

    when I integrate this last integral I get this

    <br />
\begin{array}{l}<br />
  = \left[ {\frac{{u^{\frac{1}{2}} }}{{\frac{1}{2}}}} \right]_e^{e^4 }  \\ <br />
  = \left[ {2u^{\frac{1}{2}} } \right]_e^{e^4 }  \\ <br />
  = \left[ {2(\ln x)^{\frac{1}{2}} } \right]_e^{e^4 }  \\ <br />
  = 2(\ln e^4 )^{\frac{1}{2}}  - 2\bold{\color{red}{(\ln e)^{\frac{1}{2}}} } \\ <br />
  = 4 - 1 \\ <br />
  = 3 \\ <br />
 \end{array}<br />

    The answer is given as 2, I'm not sure what I have done wrong here ?
    You used the somehow rare property:

    1^{\frac12} = \dfrac12
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  3. #3
    Member
    Joined
    Jun 2008
    Posts
    175
    doh. lol Silly arithmetic mistakes

    Thanks
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