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Name for integration by parts shortcut

  1. Oct 13, 2009 #1
    Hi all. I've recently learned a shortcut for integration by parts, but don't know what it's called or where it comes from.

    The trick is to find [tex]\lambda[/tex] such that [tex]f'' = \lambda f[/tex] and [tex]\mu[/tex] such that [tex]g'' = \mu g[/tex], providing both are constants and [tex]\lambda[/tex][tex]\neq[/tex][tex]\mu[/tex]. Then [tex]\int[/tex]f(x)g(x)dx = [tex]\frac{f'g-fg'}{\lambda-\mu}[/tex].

    Can anyone tell me what this is called? Thanks.
     
  2. jcsd
  3. Oct 13, 2009 #2

    HallsofIvy

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    Is [itex]\lambda[/itex] a number here? Then I can't imagine that "shortcut" being much good. It is only possible to find [itex]\lambda[/itex] such that [itex]f"= \lambda f[/itex] and [itex]g"= \lambda g[/itex] when both f and g are linear combinations of [itex]e^{\lamba x}[/itex] and [itex]e^{-\lambda x}[/itex].
     
    Last edited: Oct 14, 2009
  4. Oct 13, 2009 #3
    Yes, both [tex]\lambda[/tex] and [tex]\mu[/tex] are numbers. Obviously this only works in cases where they exist, such as for exponentials, trig functions, or x. But these make up a great many common integrals, so it's a pretty good shortcut.
     
  5. Oct 14, 2009 #4
    It does not seem to be a shortcut but rather like a nice exam question that my professors used to ask. I really liked it. Let me write it again, becaues latex parser went crazy above...

    [tex] fg = (\lambda -\mu)\frac{fg}{\lambda -\mu} = \frac{f''g-fg''}{\lambda - \mu} = \frac{f''g-fg''+f'g' - f'g'}{\lambda - \mu} = \frac{(f'g)' -(fg')'}{\lambda - \mu}[/tex]

    Integration gives the result...
     
  6. Oct 14, 2009 #5

    HallsofIvy

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    It would seem to me that integrals of the form [itex]\int(Ae^{\lambda x}+ Be^{-\lamba x})(Ce^{\lambda x}+ De^{-\lambda x}) dx[/itex][itex]= \int (ACe^{2\lambda x}+ BDe^{-2\lambda x}+ (BC+ AD))dx[/itex] could be done directly without worrying about integration by parts.
     
  7. Oct 14, 2009 #6
    Damn, I could have done that! I need to stop being so lazy.

    Thanks.
     
  8. Oct 14, 2009 #7
    It's for integrals like [itex]\int x sin(x) dx[/itex], which are usually done by parts. Not that this is super hard, but with the shortcut, [tex]\lambda=0, \mu = -1[/tex], and the result is [tex]sin(x)-xcos(x)[/tex] and we're done.


    PS, my apologies to any readers put off by the formatting. I'm utterly unfamiliar with this tex style and working through it as I go.
     
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