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

#### Rachael_Victoria

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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#### quasar987

Homework Helper
Gold Member
Well

$$\int_{a}^{\infty} = \lim_{M\rightarrow \infty}\int_{a}^{M}$$

So I guess you can use the result from your table and evaluate the limit.

#### ColdFusion85

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.

#### xanthym

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

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#### Rachael_Victoria

Cool thanks everyone.

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