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

  • Thread starter stlukits
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  • #1
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Homework Statement



I need to integrate an expression of the form

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

Homework Equations



I know that

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

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?)
 

Answers and Replies

  • #2
SammyS
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Homework Statement



I need to integrate an expression of the form [itex]e^{ax^{2}+bx+c}[/itex] ← [ itex]e^{ax^{2}+bx+c}[ /itex]

Homework Equations



I know that [itex]\displaystyle \int_{a}^{b}e^{-y^{2}}dy=\frac{\sqrt{\pi}}{2}(\mbox{erf}(b)-\mbox{erf}(a))[/itex] ← [ itex]\displaystyle \int_{a}^{b}e^{-y^{2}}dy=

\frac{\sqrt{\pi}}{2}(\mbox{erf}(b)-\mbox{erf}(a))[ /itex]


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?)
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:
  • #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.
 
  • #4
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Great help. Let me try it and see where it goes.
 
  • #5
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Thank you, jambaugh. It worked beautifully.
 
  • #6
jambaugh
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Thank you, jambaugh. It worked beautifully.
You're welcome, glad it worked out well.
 

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