1. Limited time only! Sign up for a free 30min personal 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!

Integration reduction formula

  1. Sep 6, 2004 #1
    I'm trying to find an integration reduction formula for the following equation:

    [tex]
    {{I}_n}=\int _{0}^{2}{{\big(4-{x^2}\big)}^n}\delta x
    [/tex]

    Any indication on how to begin would be much appreciated as I've tried many different approaches but all have ended in failure.

    Thanks
     
    Last edited: Sep 6, 2004
  2. jcsd
  3. Sep 6, 2004 #2

    Tide

    User Avatar
    Science Advisor
    Homework Helper

    I'd try to repeatedly integrate by parts or possibly use the binomial expansion.
     
  4. Sep 6, 2004 #3

    Zurtex

    User Avatar
    Science Advisor
    Homework Helper

    I think I have a solution just give me 5 mins to see if it works.
     
  5. Sep 6, 2004 #4

    Zurtex

    User Avatar
    Science Advisor
    Homework Helper

    O.K it's been quite a few months since I've done this, so I can't remember if this reduction formulae is fairly simple.

    If you use the substitution:

    [tex]x = 2 \sin u[/tex]

    It becomes:

    [tex]\frac{1}{2}4^n\int_0^{\frac{\pi}{2}} \left( \cos^{2n-1} u \right) du[/tex]

    I'm sure that can be done with a few trig identities and standard results but it's too late for me to think about it sorry.
     
  6. Sep 6, 2004 #5
    Thanks for the swift responses guys I'll have a go at that tomorrow.
     
  7. Sep 7, 2004 #6

    Zurtex

    User Avatar
    Science Advisor
    Homework Helper

    I'm not sure about this but referring back to my previous post could you just let m = 2n - 1 for n > 0 and then that's a fairly standard reduction formulae. I've never done something like that for a reduction formulae but I don't see why it can't be done.
     
  8. Sep 7, 2004 #7
    You may well be able to do that, Zurtex, and I also don't see why it wouldn't work, the only trouble is that it wouldn't prove the relation I was asked to prove.

    I managed to solve it (with help from maths teacher) using a very clever trick indeed. The solution is as follows if anyone is interested:

    [tex]
    {I }_n}\multsp =\int _{0}^{2}{{\big(4-{x^2}\big)}^n}\delta x \\\noalign\vspace{1.08333ex}} \\= {{{{\big[x{{\big(4-{x^2}\big)}^n}\big]}_0}}^2}+2n\int _{0}^{2}{x^2}{{\big(4-{x^2}\big)}^{n-1}}\delta
    x \\\noalign{\vspace{1.08333ex}} \\ \multsp \multsp \multsp \multsp \multsp \multsp =\multsp 2n\int _{0}^{2}\big(4-\big[4-{x^2}\big]\big){{\big(4-{x^2}\big)}^{n-1}}\delta x
    [/tex]

    [tex]
    \noalign{\vspace{1.08333ex}} \\ {{I }_n}\multsp \multsp =\multsp 8n\int _{0}^{2}{{\big(4-{x^2}\big)}^{n-1}}\delta x-2n\multsp {{I }_n} \\ \noalign{\vspace{0.833333ex}}
    [/tex]

    [tex]
    {{I }_n}\multsp \multsp =\multsp 8n\multsp {{I }_{n-1}}-2n\multsp {{I }_n}
    [/tex]

    [tex]
    \noalign{\vspace{0.916667ex}} \\
    {{I }_n}\multsp \multsp =\multsp \frac{8n}{2n+1}{{I }_{n-1}
    [/tex]

    The trick, which I wouldn't have thought of for a very long time, was to write the [tex]x^2[/tex] term as [tex](4-[4-x^2])[/tex]
     
    Last edited: Sep 7, 2004
  9. Sep 7, 2004 #8

    Zurtex

    User Avatar
    Science Advisor
    Homework Helper

    I thought I'd seen that before, that's really silly of me not to spot. Well done for working it out.
     
  10. Apr 1, 2009 #9
    What happens in the very first step of the solution?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Integration reduction formula
  1. Percentage reduction (Replies: 10)

Loading...