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Integration problem

  1. Mar 28, 2010 #1
    The problem statement, all variables and given/known data
    Use appropriate substitution and than a trigonometric substitution and evaluate the integral.

    [tex]\int_{1}^{e}\frac{dy}{y\sqrt{1 + (lny)^{2}}}[/tex]


    The attempt at a solution

    [tex]\int_{1}^{e}\frac{dy}{y\sqrt{1 + (lny)^{2}}}[/tex]

    [tex]ln y = tan\theta[/tex]
    [tex]y = cos^{2}\theta[/tex]
    [tex]dy = -2cos\theta sin\theta d\theta[/tex]

    [tex]= -2\int_{1}^{e}\frac{cos\theta sin\theta d\theta}{cos^{2}\theta\sqrt{1 + tan^{2}\theta}}[/tex]

    [tex]= -2\int_{1}^{e}\frac{sin\theta d\theta}{cos\theta sec\theta}[/tex]

    [tex]= -2\int_{1}^{e}sin\theta d\theta[/tex]


    How do I proceed from here? I think I have to change the limits of integration in terms of [tex]\theta[/tex] instead of [tex]y[/tex].
     
    Last edited: Mar 28, 2010
  2. jcsd
  3. Mar 28, 2010 #2

    rock.freak667

    User Avatar
    Homework Helper

    From lny=tanθ, you should get that dy/y =sec2θ dθ

    giving you


    [tex]\int \frac{sec^2\theta}{\sqrt{1+tan^2 \theta}}d\theta[/tex]
     
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