Conquering the Integral of (1/x)*exp(-ax^2): A Scientific Inquiry

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  • #1
GrandsonOfMagnusCarl
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Homework Statement
Essentially - find <1/x>, i.e. the mean of 1/x. The distribution probability density is of the form exp(-ax^2).
Relevant Equations
Mean of G = integrate ( G f(x) ) dx
Hopeless. I tried to use Taylor expansion but the zeroes and infinities go out of control really quick.
I tried WolframAlpha and it gave a special function.
What integrating trick am I missing? Or is it nonsense to solve it simply by hand?
 
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  • #2
The expectation value is infinite.
 
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Orodruin said:
The expectation value is infinite.
Ah I'm so stupid. Thank you. Also another reality check for me.
 
  • #4
<1/x>=0 as 1/x is an odd function and G*f(x) is then also odd.
 

1. What is the purpose of integrating (1/x)*exp(-ax^2)?

The purpose of integrating (1/x)*exp(-ax^2) is to find the area under the curve of the function. This type of integration is commonly used in statistics and physics to calculate probabilities and solve differential equations.

2. How do you solve the integral of (1/x)*exp(-ax^2)?

To solve the integral of (1/x)*exp(-ax^2), you can use the substitution method by letting u = -ax^2. This will transform the integral into a standard form that can be solved using basic integration techniques.

3. What is the significance of the constant "a" in the function (1/x)*exp(-ax^2)?

The constant "a" in the function (1/x)*exp(-ax^2) represents the rate at which the function decays. It affects the shape and behavior of the curve, and can also determine the convergence or divergence of the integral.

4. Can (1/x)*exp(-ax^2) be integrated using other methods besides substitution?

Yes, (1/x)*exp(-ax^2) can also be integrated using other methods such as integration by parts or partial fractions. However, the substitution method is the most commonly used and efficient method for this type of integral.

5. What are the applications of integrating (1/x)*exp(-ax^2)?

Integrating (1/x)*exp(-ax^2) has various applications in mathematics, physics, and engineering. It is used to solve differential equations, calculate probabilities in statistics, and analyze the behavior of systems in physics. It is also used in signal processing and image processing algorithms.

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