Help with a tricky improper integration problem

  • #1
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0
Show that:

∫(x^2)e^(-x^2)=(1/2)∫(e^(-x^2)

(both integrals are from 0 to ∞)



I don't even know if it's possible to integrate these functions, so it makes me think that what's interesting about the problem is that it only works with this integration limits.

I know improper integration, but don't know if prooving this "numerically" is the right path.

Thanks, I would appreciate any advice
 

Answers and Replies

  • #2
Curious3141
Homework Helper
2,843
87
Show that:

∫(x^2)e^(-x^2)=(1/2)∫(e^(-x^2)

(both integrals are from 0 to ∞)



I don't even know if it's possible to integrate these functions, so it makes me think that what's interesting about the problem is that it only works with this integration limits.

I know improper integration, but don't know if prooving this "numerically" is the right path.

Thanks, I would appreciate any advice
Hint:

[tex]\int_0^{\infty} x^2e^{-x^2}dx = \int_0^{\infty} x.(xe^{-x^2})dx[/tex]

Integrate that by parts (use [itex]u = x, dv = xe^{-x^2}dx[/itex]), then use L'Hopital's Rule to calculate the limit of one of the expressions.
 
Last edited:
  • #3
107
0
Hint:

[tex]\int_0^{\infty} x^2e^{-x^2}dx = \int_0^{\infty} x.(xe^{-x^2})dx[/tex]

Integrate that by parts (use [itex]u = x, dv = xe^{-x^2}dx[/itex]), then use L'Hopital's Rule to calculate the limit of one of the expressions.
is this correct?

I integrated by pats (differentiating x and integrating x^2e^{-x^2} )

until I got to the point where (e^{-x^2})/2 (evaluated from o to infinity)=0

Using L'Hopital's Rule, I effectively got 0=0




Is this the way?
 
  • #4
Curious3141
Homework Helper
2,843
87
is this correct?

I integrated by pats (differentiating x and integrating x^2e^{-x^2} )

until I got to the point where (e^{-x^2})/2 (evaluated from o to infinity)=0

Using L'Hopital's Rule, I effectively got 0=0




Is this the way?
No, I think you messed up the integration by parts. You should end up with:

[tex]\int_0^{\infty} x^2e^{-x^2}dx = xe^{-x^2}|_0^{\infty} +\frac{1}{2}\int_0^{\infty}e^{-x^2}dx[/tex]

and the first expression on the RHS vanishes when you take the limit by L' Hopital's.
 
  • #5
107
0
No, I think you messed up the integration by parts. You should end up with:

[tex]\int_0^{\infty} x^2e^{-x^2}dx = xe^{-x^2}|_0^{\infty} +\frac{1}{2}\int_0^{\infty}e^{-x^2}dx[/tex]

and the first expression on the RHS vanishes when you take the limit by L' Hopital's.
It's the same thing (maybe my english isn't that good, I don't know) but Thanks!!!
 
  • #6
Curious3141
Homework Helper
2,843
87
It's the same thing (maybe my english isn't that good, I don't know) but Thanks!!!
How is it the same thing? You said you got "0 = 0" which means nothing. You also said you got [itex]\frac{1}{2}\int_0^{\infty}e^{-x^2}dx = 0[/itex] which is clearly wrong.

Anyway, if you're confident you have the right solution, then it's fine.
 
  • #7
107
0
What about this one? Show:

∫e-x2=∫√(-lny)
(from 0 to ∞) (from 0 to 1)



I tried this one by solving for y in the right side of the equation, getting y=-ex2

However, when checking the solution graphically, it's not right.

So what the hell am I supposed to do? I also wonder if the integration limits changed when expressed in y or x.


Thanks
 
  • #8
296
0
You should do a substitution to change the limits and get the log into play, then integrate by parts. Your previous result might help here.
 
Last edited:
  • #9
Curious3141
Homework Helper
2,843
87
What about this one? Show:

∫e-x2=∫√(-lny)
(from 0 to ∞) (from 0 to 1)



I tried this one by solving for y in the right side of the equation, getting y=-ex2

However, when checking the solution graphically, it's not right.

So what the hell am I supposed to do? I also wonder if the integration limits changed when expressed in y or x.


Thanks
Use this (your previous result):

[tex]\int_0^{\infty} x^2e^{-x^2}dx = \frac{1}{2}\int_0^{\infty}e^{-x^2}dx[/tex]

and make the substitution [itex]e^{-x^2} = y[/itex] on the LHS. The result follows very quickly.
 

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