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Integral of exp-(ax^2+bx+c)

  1. Jul 12, 2011 #1
    1. The problem statement, all variables and given/known data

    I need to integrate an expression of the form

    [tex]e^{ax^{2}+bx+c}[/tex]

    2. Relevant equations

    I know that

    [tex]\int_{a}^{b}e^{-y^{2}}dy=\frac{\sqrt{\pi}}{2}(\mbox{erf}(b)-\mbox{erf}(a))[/tex]

    3. The attempt at a solution

    I tried to substitute [tex]ax^{2}+bx+c[/tex] by [tex]-y^{2}[/tex] but I get hopelessly tangled. (PS.: how do get the tex tags to not create an equation environment but stay inline?)
     
  2. jcsd
  3. Jul 12, 2011 #2

    SammyS

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    Use itex & /itex for inline LATEX. Use \displaystyle with that to keep , fractions, etc. full size. See above.

    Try completing the square for [itex]ax^{2}+bx+c[/itex] .
     
    Last edited: Jul 12, 2011
  4. Jul 12, 2011 #3

    jambaugh

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    First you need to complete the square on the quadratic. Note that the quadratic formula is derived this way so basically encodes it:
    [itex] f(x) = ax^2 + b x + c = a(x-h)^2 + k[/itex]
    where [itex]h = -b/2a[/itex] and [itex]k = f(h)=ah^2 + bh + c[/itex].

    Second, note that the constant term in the exponent can be factored out:
    [itex] e^{a(x-h)^2 + k} = e^{a(x-h)^2}e^k[/itex]

    See where that gets you.
     
  5. Jul 12, 2011 #4
    Great help. Let me try it and see where it goes.
     
  6. Jul 14, 2011 #5
    Thank you, jambaugh. It worked beautifully.
     
  7. Jul 14, 2011 #6

    jambaugh

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    You're welcome, glad it worked out well.
     
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