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

  • Thread starter noowutah
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In summary, to integrate an expression of the form e^{ax^{2}+bx+c}, complete the square on the quadratic and factor out the constant term in the exponent. This will make it easier to integrate.
  • #1
noowutah
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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?)
 
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  • #2
stlukits said:

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] .
 
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  • #3
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
Great help. Let me try it and see where it goes.
 
  • #5
Thank you, jambaugh. It worked beautifully.
 
  • #6
stlukits said:
Thank you, jambaugh. It worked beautifully.
You're welcome, glad it worked out well.
 

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

What is the formula for the integral of exp-(ax^2+bx+c)?

The formula for the integral of exp-(ax^2+bx+c) is ∫e^-(ax^2+bx+c)dx = √(π/a)e^(c-b^2/4a)erf(√a(x+b/2a)), where erf is the error function.

How do you solve for the integral of exp-(ax^2+bx+c)?

To solve for the integral of exp-(ax^2+bx+c), you can use the substitution method or complete the square to rewrite the expression as a standard form of the Gaussian integral. Then, apply the appropriate formula to solve for the integral.

Can the integral of exp-(ax^2+bx+c) be evaluated using basic calculus techniques?

No, the integral of exp-(ax^2+bx+c) cannot be evaluated using basic calculus techniques. It requires the use of special functions such as the error function and the Gaussian integral formula.

What is the purpose of finding the integral of exp-(ax^2+bx+c)?

Finding the integral of exp-(ax^2+bx+c) is useful in various applications such as probability and statistics, quantum mechanics, and signal processing. It allows us to determine the area under the curve of a Gaussian function, which is a common distribution in many fields.

Are there any real-life examples where the integral of exp-(ax^2+bx+c) is used?

Yes, the integral of exp-(ax^2+bx+c) is used in various real-life examples such as calculating the probability of a normally distributed event, determining the energy distribution of a quantum particle, and filtering signals in communication systems.

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