Results 1 to 2 of 2

Thread: trig integral

  1. #1
    mms
    mms is offline
    Junior Member
    Joined
    Jul 2009
    Posts
    67

    trig integral

    prove that
    $\displaystyle
    \int\limits_0^{2\pi } {\frac{{xsen(x)}}
    {{1 + \cos ^2 (x)}}dx = } \frac{\pi }
    {2}\int\limits_{ - 1}^1 {\frac{1}
    {{1 + x^2 }}dx}

    $

    and calculate the integral
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Jan 2009
    Posts
    715
    $\displaystyle \int_0^{\pi} \frac{x\sin(x)}{1 + \cos^2(x)}~dx $

    Sub. $\displaystyle x = \pi - t$

    $\displaystyle dx = -dt $

    $\displaystyle I = \int_0^{\pi} \frac{(\pi-t)\sin(\pi - t)}{1 + \cos^2(\pi -t)}~dt $

    $\displaystyle I = \int_0^{\pi} \pi \cdot \frac{\sin(t)}{1 + \cos^2(t)}~dt - I$

    $\displaystyle \implies $

    $\displaystyle 2I = \pi \int_0^{\pi} \frac{\sin(t)}{1 + \cos^2(t)} ~dt$

    Sub $\displaystyle \cos(t) = u$ $\displaystyle \sin(t) dt = -du $

    $\displaystyle 2I = \pi \int_{-1}^1 \frac{1}{1 + u^2}~du$

    $\displaystyle I = \frac{\pi}{2} \int_{-1}^1 \frac{1}{1 + u^2}~du $

    $\displaystyle I = (\frac{\pi}{2})^2 $
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Trig Integral
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Dec 1st 2010, 06:09 PM
  2. Need help with Trig Integral
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Feb 23rd 2010, 02:39 AM
  3. trig integral
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Apr 11th 2009, 02:26 PM
  4. a trig integral
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Aug 30th 2008, 02:56 PM
  5. trig integral
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Mar 7th 2006, 04:12 AM

Search Tags


/mathhelpforum @mathhelpforum