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

1. Jul 12, 2011

stlukits

1. The problem statement, all variables and given/known data

I need to integrate an expression of the form

$$e^{ax^{2}+bx+c}$$

2. Relevant equations

I know that

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

3. The attempt at a solution

I tried to substitute $$ax^{2}+bx+c$$ by $$-y^{2}$$ but I get hopelessly tangled. (PS.: how do get the tex tags to not create an equation environment but stay inline?)

2. Jul 12, 2011

SammyS

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

Try completing the square for $ax^{2}+bx+c$ .

Last edited: Jul 12, 2011
3. Jul 12, 2011

jambaugh

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

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

See where that gets you.

4. Jul 12, 2011

stlukits

Great help. Let me try it and see where it goes.

5. Jul 14, 2011

stlukits

Thank you, jambaugh. It worked beautifully.

6. Jul 14, 2011

jambaugh

You're welcome, glad it worked out well.

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