Results 1 to 2 of 2

Math Help - Prove a simple reduction formula

  1. #1
    Newbie
    Joined
    Feb 2013
    From
    australia
    Posts
    1

    Prove a simple reduction formula

    How would I prove the following reduction formula?

    ∫dx/((x^2)*(sqrt(a^2 - x^2))=-sqrt(a^2 - x^2)/(a^2 x)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,660
    Thanks
    600

    Re: Prove a simple reduction formula

    Hello, harvey123!

    This is not a Reduction Formula . . . it is just an integration.


    \int\frac{dx}{x^2\sqrt{a^2-x^2}} \;=\;-\frac{\sqrt{a^2-x^2}}{a^2x} + C

    Let x \,=\,a\sin\theta \quad\Rightarrow\quad dx \,=\,a\cos\theta\,d\theta

    Substitute: . \int \frac{a\cos\theta\,d\theta}{a^2\sin^2\theta\!\cdot  \!a\cos\theta} \;=\;\frac{1}{a^2}\int\csc^2\theta\,d\theta \;=\; -\frac{1}{a^2}\cot\theta + C


    Back-substitute: . \sin\theta \,=\,\frac{x}{a} \,=\,\frac{opp}{hyp} \quad\Rightarrow\quad adj \,=\,\sqrt{a^2-x^2}

    . . . . . . . . . . . . . \cot\theta \,=\,\frac{adj}{opp} \,=\,\frac{\sqrt{a^2-x^2}}{x}


    Answer: . -\frac{1}{a^2}\cdot\frac{\sqrt{a^2-x^2}}{x}+C \;=\;-\frac{\sqrt{a^2-x^2}}{a^2x} + C
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: February 23rd 2010, 09:05 AM
  2. Replies: 5
    Last Post: February 19th 2010, 05:37 PM
  3. reduction formula.
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 10th 2009, 03:23 AM
  4. Reduction formula
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 2nd 2008, 10:22 PM
  5. Reduction Formula
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 22nd 2008, 05:00 PM

Search Tags


/mathhelpforum @mathhelpforum