Integration of X^2 exp(-aX^2) from 0 to infinity

In summary, you can use the result from your table and evaluate the limit. You are correct in that you have to use Integration By Parts. However, since the upper limit is infinity, you have to use the method for solving infinite limits of integration. Hope this helps!
  • #1
Rachael_Victoria
16
0
Hey can someone tell me the value of the integral of X^2 exp (-aX^2) dx from zero to infinity. I have the general solution from a table of integrals but since the upper limit is infinity, I can't really plug these numbers in. Can't find it in a table of integrals anywhere? If anyone has the answer it would be really great as I could then continue on in my P-chem homework. I also know how to do this with integration by parts, but there is the whole catch of infinity as the upper limit, anything beyond doing integration is past my personal education and capabilities.
Thanks
Rachael
 
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  • #2
Well

[tex]\int_{a}^{\infty} = \lim_{M\rightarrow \infty}\int_{a}^{M}[/tex]

So I guess you can use the result from your table and evaluate the limit.
 
  • #3
You are correct in that you have to use Integration By Parts. However, since the upper limit is infinity, you have to use the method for solving infinite limits of integration. In your case, change the upper limit (infinity) to b , then solve the integral. Then you simply take the limit of the result as b goes to infinity (you will have an expression in terms of b since you replaced the upper limit by b). I haven't worked out the problem, but depending on the functions involved, the integral can either converge or diverge (i.e. it would diverge if you had something like lim as b goes to infinity of[1-cos(b)] ). I am guessing in your case if it is required to solve subsequent problems in your homework that it will converge. Hope that helps.
 
  • #4
Rachael_Victoria said:
Hey can someone tell me the value of the integral of X^2 exp (-aX^2) dx from zero to infinity. I have the general solution from a table of integrals but since the upper limit is infinity, I can't really plug these numbers in. Can't find it in a table of integrals anywhere? If anyone has the answer it would be really great as I could then continue on in my P-chem homework. I also know how to do this with integration by parts, but there is the whole catch of infinity as the upper limit, anything beyond doing integration is past my personal education and capabilities.
Thanks
Rachael

[tex] 1: \ \ \ \ \int_{0}^{\infty} x^{2} \cdot \exp(-ax^{2}) \, dx \ \ = \ \ \frac{\sqrt{\pi}} {4 \cdot a^{3/2}} [/tex]


~~
 
  • #5
Cool thanks everyone.
 

1. What is the purpose of integrating X^2 exp(-aX^2) from 0 to infinity?

The purpose of integrating X^2 exp(-aX^2) from 0 to infinity is to find the area under the curve of the function. This is often used in mathematical and scientific calculations to determine the probability of certain events or to find the average value of a variable.

2. Why is the upper limit of integration set to infinity?

The upper limit of integration is set to infinity because the function X^2 exp(-aX^2) does not have a finite endpoint. This means that the curve continues infinitely in the positive X direction and it is necessary to integrate to infinity in order to accurately capture the entire area under the curve.

3. How is the value of a determined in the integration?

The value of a is determined through experimentation or data analysis. In scientific studies, a is often a variable that is being studied and its value is determined through careful measurement and analysis. In mathematical calculations, a may be chosen based on certain criteria or constraints set by the problem at hand.

4. What is the significance of the term X^2 in the function?

The term X^2 in the function is a polynomial term that has a significant impact on the shape of the curve. It causes the function to increase rapidly and then decrease rapidly, resulting in a bell-shaped curve. This shape is often seen in natural phenomena and is known as a Gaussian distribution.

5. Are there any practical applications of integrating X^2 exp(-aX^2) from 0 to infinity?

Yes, there are several practical applications of integrating X^2 exp(-aX^2) from 0 to infinity. This function is commonly used in statistics to determine the probability of a particular event occurring. It is also used in physics and engineering to analyze and model natural phenomena. Additionally, it is used in various fields of science and mathematics to solve complex problems and make predictions.

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