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Why subsitution method for integration always work ?

  1. Oct 27, 2011 #1

    MIB

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    why substitution method for integration always work ?

    Why can we completely treat dx and du known in substitution method completely like differentials even if we don't have ∫f(g(x))g'(x) dx , i.e : why we can substitute x in terms of u and dx in terms of du .

    thanks
     
    Last edited: Oct 27, 2011
  2. jcsd
  3. Oct 28, 2011 #2

    lurflurf

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    Homework Helper

    The chain rule and implicit function theorem
    if g is chosen as a function of x we have
    f(g(x))g'(x) dx=f(g)dg
    which comes from
    dg=g'(x)dx
    which is the chain rule precisely
    if instead we chose an implicit relationship between g and x we have
    h(g,x)=0
    dh=hxdx+hgdg=0
    dg=[dg/dx]dx=[-hx/hg]dx
    which is the chain rule precisely
     
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