How can I integrate e^{ax^{2}+bx+c} without getting tangled?

[noowutah]

Homework Statement

I need to integrate an expression of the form

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

Homework Equations

I know that

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

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?)

 

[SammyS]

Homework Statement

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

Homework Equations

I know that $\displaystyle \int_{a}^{b}e^{-y^{2}}dy=\frac{\sqrt{\pi}}{2}(\mbox{erf}(b)-\mbox{erf}(a))$ ← [ 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 $$ax^{2}+bx+c$$ by $$-y^{2}$$ but I get hopelessly tangled. (PS.: how do get the tex tags to not create an equation environment but stay inline?)

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

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

 

[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.

 

[noowutah]

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

 

[noowutah]

Thank you, jambaugh. It worked beautifully.

 

[jambaugh]

Thank you, jambaugh. It worked beautifully.

You're welcome, glad it worked out well.