Results 1 to 6 of 6

Math Help - integral

  1. #1
    Member
    Joined
    Dec 2009
    Posts
    79

    integral

    find :
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Sep 2009
    Posts
    177
    Thanks
    1
    What have you tried so far, have you tried any identities?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,318
    Thanks
    1234
    Hint:

    \displaystyle \cos{(n\theta)} \equiv \sum_{k=0}^{n}{n\choose{k}}\cos^k{\theta}\sin^{n-k}{\theta}\cos{\left[\frac{1}{2}(n-k)\pi\right]}
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Dec 2009
    Posts
    79
    Yes, I understand
    By
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Mar 2010
    Posts
    715
    Thanks
    2
    Either using \int_{a}^{b}f(x)\;{dx} = \int_{a}^{b}f(a+b-x)\;{dx} or putting x \mapsto x-\pi gives:

    \displaystyle \begin{aligned} I & = \int_{0}^{\pi} \cos{x}\cos{3x}\cos{5x} \\& = \int_{0}^{\pi} \cos(\pi-x)\cos(3\pi-3x)\cos(5\pi-5x) \\& = -\int_{0}^{\pi} \cos{x}\cos{3x}\cos{5x}. \end{aligned}

    Hence 2I = 0, thus I = 0.

    More generally, if n is even, then:

    \displaystyle  \begin{aligned} & \int_{0}^{\pi}\prod_{0\le k \le n}\cos\left[(2k+1)x\right]\;{dx}  \\& = \int_{0}^{\pi}\prod_{0\le k \le n}\cos\left[(2k+1)\pi-(2k+1)x\right]\;{dx} \\& = -\int_{0}^{\pi}\prod_{0\le k \le n}\cos\left[(2k+1)x\right]\;{dx}. \end{aligned}

    Thus \displaystyle \int_{0}^{\pi}\prod_{0\le k \le n}\cos[(2k+1)x]\;{dx} = 0.
    Last edited by TheCoffeeMachine; February 12th 2011 at 01:15 AM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,654
    Thanks
    12
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: August 31st 2010, 07:38 AM
  2. Replies: 1
    Last Post: June 2nd 2010, 02:25 AM
  3. Replies: 0
    Last Post: May 9th 2010, 01:52 PM
  4. [SOLVED] Line integral, Cauchy's integral formula
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: September 16th 2009, 11:50 AM
  5. Replies: 0
    Last Post: September 10th 2008, 07:53 PM

Search Tags


/mathhelpforum @mathhelpforum