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Thread: Can't find the double integral of ln function?

  1. #1
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    Can't find the double integral of ln function?

    Can't find the double integral of ln function?-math.png

    i am taking the outer integral from 0 to 1 for dx , inner is e^x to e^sqrtx. Kinda stuck on how to solve the integral. I do not have many answer submissions left . Integration calculator doesn't know how to evaluate it either.
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  2. #2
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    Re: Can't find the double integral of ln function?

    $\Large \displaystyle \int_0^1 \int_{\frac{e^x}{12}}^\frac{e^{\sqrt{x}}}{12} ~\dfrac{1}{\ln(12y)}~dy~dx$

    Trying to integrate it in this order is a mess. a chocolate mess.

    So let's flip the order of integration, especially as $f(x,y)$ doesn't depend on $x$ at all

    $\Large \displaystyle \int_{\frac{1}{12}}^{\frac{e^1}{12}} \int_{\ln(12y)^2}^{\ln(12y)}~\dfrac{1}{\ln(12y)}~d x~dy = $

    $\Large \displaystyle \int_{\frac{1}{12}}^{\frac{e^1}{12}}~\dfrac{\ln (12 y)-\ln ^2(12 y)}{\ln (12 y)}~dy =$

    $\Large \displaystyle \int_{\frac{1}{12}}^{\frac{e^1}{12}}~1 - \ln(12y)~dy =$

    $\Large \left . 2 y-y \log (12 y) \right|_\frac{1}{12}^{\frac{e^1}{12}} = \dfrac{1}{12} (e-2) \approx 0.0598568$
    Thanks from topsquark and HallsofIvy
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