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Integration problem

  • Thread starter Precursor
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  • #1
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Homework Statement
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:

Answers and Replies

  • #2
rock.freak667
Homework Helper
6,230
31
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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