Results 1 to 6 of 6

Math Help - a problem about integral

  1. #1
    Newbie
    Joined
    Nov 2011
    Posts
    21

    a problem about integral


    could you help me please. any help will be appreciated...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Siron's Avatar
    Joined
    Jul 2011
    From
    Norway
    Posts
    1,250
    Thanks
    20

    Re: a problem about integral

    Have you tried with induction? (Maybe that will work)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: a problem about integral

    Quote Originally Posted by mami View Post
    could you help me please. any help will be appreciated...
    Could you help us mentioning the subject where that integral is included? For example, sometimes we solve an integral using Complex Variable and we receive the following answer: Sorry, we have not covered Complex Variable, etc., etc., etc.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: a problem about integral

    I have become interested in this problem and have spent a great deal of time trying to get induction (among other things) to work, but have not been successful. Hopefully, someone can show me where I am going wrong. Obviously the statement is true for the base case of n = 1. So I stated the induction hypothesis P_n:

    \int_0^{\pi}\frac{\cos(nx)-\cos(na)}{\cos(x)-\cos(a)}\,dx=\pi\frac{\sin(na)}{\sin(a)}

    Assuming this is true, I then added P_{n+2} to get:

    \int_0^{\pi}\frac{\cos(nx)-\cos(na)}{\cos(x)-\cos(a)}\,dx+\int_0^{\pi}\frac{\cos\left((n+2)x \right)-\cos\left((n+2)a \right)}{\cos(x)-\cos(a)}\,dx=

    \pi\frac{\sin(na)}{\sin(a)}+\pi\frac{\sin\left((n+  2)a\right)}{\sin(a)}

    \int_0^{\pi}\frac{\left(\cos\left((n+2)x \right)+\cos(nx)\right)-\left(\cos\left((n+2)a\right)+\cos(na) \right)}{\cos(x)-\cos(a)}\,dx=\frac{\pi}{\sin(a)}\left(\sin\left((n  +2)a\right)+\sin(na)\right)

    \int_0^{\pi}\frac{\left(2\cos\left((n+1)x \right)\cos(x)\right)-\left(2\cos\left((n+1)a\right)\cos(a)\right)}{\cos  (x)-\cos(a)}\,dx=\frac{\pi}{\sin(a)}\(2\sin\left((n+1)  a\right)\cos(a)\)

    \int_0^{\pi}\frac{\cos\left((n+1)x\right) \frac{\cos(x)}{\cos(a)}-\cos\left((n+1)a\right)}{\cos(x)-\cos(a)}\,dx=\pi\frac{\sin\left((n+1)a \right)}{\sin(a)}

    I can't figure out how to get rid of the \frac{\cos(x)}{\cos(a)} in the numerator of the integrand.

    edit: sorry about my edits...this board has some unexpected idiosyncrasies concerning the
    \left and \right operators in \LaTeX...
    Last edited by MarkFL; January 2nd 2012 at 11:53 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: a problem about integral

    I should add that this problem has also been posted on the forum which I help moderate, and I have posted the same problem I have in using induction.

    The OP is an infrequent visitor there, and it may be a while before we hear from them to get any feedback on the context of the problem.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: a problem about integral

    The following fix was given to me:

    \int_0^{\pi}\frac{\cos((n+1)x)}{\cos(a)}\,dx=0

    \int_0^{\pi}\frac{\cos((n+1)x)\left(\frac{\cos(x)}  {\cos(a)}-1\right)+\cos((n+1)a)-\cos((n+1)a)}{\cos(x)-\cos(a)}\,dx=0

    \int_0^{\pi}\frac{\cos((n+1)x)\frac{\cos(x)}{\cos(  a)}-\cos((n+1)a)}{\cos(x)-\cos(a)}\,dx=\int_0^{\pi}\frac{\cos((n+1)x)-\cos((n+1)a)}{\cos(x)-\cos(a)}\,dx
    Last edited by MarkFL; January 4th 2012 at 10:54 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Integral problem involving the definition of the integral
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 23rd 2011, 08:06 AM
  2. integral problem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: July 22nd 2010, 02:06 AM
  3. Integral problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 12th 2010, 02:46 PM
  4. Integral Problem
    Posted in the Calculus Forum
    Replies: 6
    Last Post: September 25th 2009, 07:11 AM
  5. Integral Problem
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: February 3rd 2009, 02:21 AM

Search Tags


/mathhelpforum @mathhelpforum