Gaussian Integral Question

In summary, the integral of (x^n)(e^(-a*x^2)) is easier to evaluate when n is odd. To evaluate the indefinite integral of x*e^(-a*x^2), a simple substitution of u = a*x^2 can be used. The resulting integral is -(1/2a)*e^(-a*x^2). When differentiating this result with respect to a, the original integrand, (x*e^(-a*x^2)), is recovered.
  • #1

Homework Statement



The integral of (x^n)(e^(-a*x^2)) is easier to evaluate when n is odd.
a) Evaluate ∫(x*e^(-a*x^2)*dx) (No computation allowed!)
b) Evaluate the indefinite integral of x*e^(-a*x^2), using a simple substitution.
c) Evaluate ∫(x*e^(-a*x^2)*dx) [from o to +inf]
d) Differentiate the previous result to evaluate ∫((x^3)(e^(-a*x^2))dx)

Homework Equations



∫(e^(-a*x^2)*dx) = (1/2)√(∏/a)


The Attempt at a Solution



I thought that the easiest solution might be using integration by parts, but I ran into the issue of the range being different on the integrals, and I have no idea how else I can do this unless I can assume that the function is even...
 
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  • #2
What are the limits on the integrals?
 
  • #3
The first one is -inf to inf, second is indefinite, third is 0 to inf, and fourth is 0 to inf.
 
  • #4
superspartan9 said:
The first one is -inf to inf, second is indefinite, third is 0 to inf, and fourth is 0 to inf.

You don't need integration by parts for any of those. Take them one at a time. Start with a). The integrand isn't even, it's odd.
 
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  • #5
Alright, I figured out the first one in class by drawing the graph for x and e^(-ax^2) and realized that the x made the function odd, as you said, and that the integral was then 0 for -inf to inf.

I'm going to try and tackle c) and d) because I have no idea where to start on b) with that substitution. My brain is stuck on integration by parts >_> if you could give me a hint or something as to where to begin with the indefinite integral, that would be great!
 
  • #6
Update: Just tried substituting u = a*x^2 into the integral, and it evaluated to -(1/2a)*e^(-a*x^2)

I think it's right... but I'm not sure, the substitution worked though because the du = 2ax dx which means we can just throw the constants in there.
 
  • #7
Another Update: The above equation is correct because it yields the correct value for part c).

Now I'm just stuck on d) where it's asking me to differentiate... More later if I figure it out.
 
  • #8
Try differentiating it and see if you recover the integrand.
 
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  • #9
Got it! Taking the derivative with respect to a of part c) yields the answer! :D
 

1. What is a Gaussian integral?

A Gaussian integral is a type of definite integral that involves an exponential function with a quadratic argument. It is named after the mathematician Carl Friedrich Gauss and is commonly used in statistics and physics.

2. How do you solve a Gaussian integral?

There are several methods for solving a Gaussian integral, including substitution, integration by parts, and completing the square. The most common approach is to use the technique of completing the square, which involves rewriting the quadratic argument in the exponential function as a perfect square.

3. What is the significance of the Gaussian integral in statistics?

The Gaussian integral is important in statistics because it is used to calculate the probability density function of normally distributed random variables. This is commonly known as the bell curve and is used to describe the distribution of many natural phenomena, such as height, weight, and IQ.

4. Can the Gaussian integral be evaluated analytically?

Yes, the Gaussian integral can be evaluated analytically using various techniques, such as the ones mentioned above. However, for some values of the parameters, the integral may not have a closed-form solution and must be approximated numerically.

5. How is the Gaussian integral related to the error function?

The Gaussian integral is closely related to the error function, which is defined as the integral of the Gaussian function from 0 to x. The error function is used in statistics to calculate the cumulative distribution function of a normal distribution, and it is also used in physics to describe the diffusion of particles.

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