1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    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

    User Avatar
    Science Advisor
    Homework Helper

    Try the substitution ##\arctan(x)=y## in ##I_1##.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Threads - Definite integral problem Date
Work problem - Rope, pulley and brick (applied integration) Nov 17, 2016
Definite integral problem Mar 24, 2015
Definite integration problem Jan 31, 2015
Definite Integral limit problems Nov 29, 2013
Definite integral problem Nov 20, 2013