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.