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Definite integral problem

  1. Sep 15, 2016 #1
    1. The problem statement, all variables and given/known data
    Hi all,

    This problem has been troubling me for a while now; even though I have tried my best ( and filled up a rough notebook in the process). Consider $$I_1=\int_{0}^{1} \frac{tan^{-1}x}{x} dx$$$, and $$I_2=\int_{0}^{\pi/2} \frac{x}{sinx}dx$$. We are supposed to find $$\frac{I_1}{I_2}$$. The answer is $$1/2$$.


    2. Relevant equations


    3. The attempt at a solution
    My try- To make the limits for both identical, I substituted $$x=sin\theta$$ in the first integral, and then tried to make use of the properties of definite integrals ( replacing $$f(\theta)$$ by $$f(\pi/2-\theta)$$ etc), but no real progress was made. I then tried $$x=arcsint$$ for the second one, but no result.

    Now I really doubt if there is something wrong with the question itself, or am I just being really silly. Please help me here. Thanks in advance!!
     
  2. jcsd
  3. Sep 15, 2016 #2

    Math_QED

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    Homework Helper

    I did not look to closely at this exercise, so I might be misleading you here.

    But, I saw, that when you substitute ##x = \arcsin t## in ##I_2##, you get:

    $$I_2 = \int\limits_0^1 \frac{arcsin t}{t\sqrt{1-t^2}}dt$$ and this integral has the same bounds as ##I_1##, so maybe you can use the linearity of the definite integral or something like that?
     
  4. Sep 15, 2016 #3

    Ray Vickson

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    Try the substitution ##\arctan(x)=y## in ##I_1##.
     
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