1. Not finding help here? Sign up for a free 30min 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!

Please help in an integration

  1. Jul 29, 2011 #1
    Please tell me what did I do wrong on this integration. The book claimed this can only be solved in numerical method and the answer is 1.218.

    [tex]\int _0^{\pi} \frac {\cos ^2\left [\frac {\pi} 2 \;\cos\;\theta\right]}{\sin \theta}\;d\;\theta [/tex]

    [tex]\hbox { Let }\; u=\frac {\pi} 2\; \cos \theta\;\Rightarrow\; d\theta =-\frac{2\;d\;u}{\pi\;\sin\;\theta}\; \Rightarrow\;\int _0^{\pi} \frac {\cos ^2\left [\frac {\pi} 2 \;\cos\;\theta\right]}{\sin \theta}\;d\;\theta \;=\; -\frac 2 {\pi}\int \cos^2\;u \;d\;u\;=-\frac 1 {\pi} \left[ \int d\;u \;+\; \int \cos(2u) d\;u \right ][/tex]

    [tex] = -\left [\frac { \frac {\pi} 2 \cos \theta}{\pi}\right]_0^{\pi} \;-\;\frac 1 {2\pi} \int \cos v \;dv \;\;\;\;\;\;\hbox { where }\;\; \;v=2u \;\hbox { and }\;d\;u=\frac {d\;v} 2 [/tex]

    [tex]= 1 -\left[\frac { \sin (2\frac {\pi} 2 \;\cos \theta)}{2\pi}\right]^{\pi}_0 \;= 1 [/tex]

    I solve this without using numerical method and the answer is 1 instead of 1.218. Who is right?
  2. jcsd
  3. Jul 29, 2011 #2
    on your first u-substitution u=pi/2*cos(x)
    your du=sin(x)dx
    your du is not [itex] du= \frac{1}{sin(x)} [/itex]
    you cant get rid of that sine in the bottom with that u substitution
  4. Jul 29, 2011 #3

    I really don't get it. Please explain more.


    Last edited: Jul 29, 2011
  5. Jul 29, 2011 #4
    yes that's what you would have. This integral looks tricky if its even doable, I tried some stuff using trig identities and stuff and even thought about using Eulers formula .
  6. Jul 29, 2011 #5
    After the substitution the integrand should become

    [itex] -\frac{2}{\pi}\frac{cos^2 u}{sin^2 \theta} [/itex]
  7. Jul 29, 2011 #6
    I spent awhile on this and eventually asked Mathematica for help in calculating the integral. It confirmed the answer as [itex]\approx 1.21883[/itex], and gave that
    [itex]\int^{\pi }_0{{cos}^2\left(\frac{\pi }{2}{\cos \left(x\right)\ }\right){\csc \left(x\right)\ }dx=\frac{1}{2}\left(-Ci(2\pi \right)+\gamma +{\rm ln}(2\pi ))}.[/itex]
    Where Ci(x) is the cosine integral (http://mathworld.wolfram.com/CosineIntegral.html) and [itex]\gamma[/itex] is the Euler-Mascheroni constant (http://mathworld.wolfram.com/Euler-MascheroniConstant.html), which would seem to confirm this as a non-elementary integral. I don't even think it's possible to solve without the aid of a very comprehensive set of integration tables. =\
    Last edited by a moderator: Apr 26, 2017
  8. Jul 29, 2011 #7
    Thanks, I was so blind!!! No wonder!!! It was late last night. I don't know why I tend to make this kind of stupid mistake all the times, it is so obvious that I miss it. That's the reason I never get 100 in my ODE class, always 96 97, always have one of these mistakes to take off a few points!!! Kicker is I still did not see it after your first reply.....and I did went through the whole thing!!!!

  9. Jul 29, 2011 #8
    Last edited by a moderator: Apr 26, 2017
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Please help in an integration