• Support PF! Buy your school textbooks, materials and every day products Here!

Help for integration

  • Thread starter Saitama
  • Start date
  • #1
3,812
92

Homework Statement


[tex]\int_{0}^{∞} x^{2n+1}.e^{-x^2}dx[/tex]
is equal to (n ε N)
a)n!
b)2(n!)
c)n!/2
d)(n+1)!/2

Homework Equations





The Attempt at a Solution


I can go on solving this by using n=1 or n=2. But i want to do it by a correct method. Is there a proper way to do it? I am having no idea, how should i begin?
 

Answers and Replies

  • #2
uart
Science Advisor
2,776
9
Hi Pranav. Make the substitution [itex]u = x^2[/itex] and see if you can manipulate it in terms of the gamma function.
 
  • #3
3,812
92
Hello uart!

Hi Pranav. Make the substitution [itex]u = x^2[/itex] and see if you can manipulate it in terms of the gamma function.
I have already tried that substitution and i end up with this:
[tex]\frac{1}{2} \int_{0}^{∞} u^n \cdot e^{-u}du[/tex]
Gamma function? I guess i have never heard of it.
Any other way to solve it? If there's no way, i would try to see what this "gamma function" is.
 
  • #4
uart
Science Advisor
2,776
9
Hello uart!


I have already tried that substitution and i end up with this:
[tex]\frac{1}{2} \int_{0}^{∞} u^n \cdot e^{-u}du[/tex]
Gamma function? I guess i have never heard of it.
Any other way to solve it? If there's no way, i would try to see what this "gamma function" is.
Yes, that result is correct. That's good as that makes it equal 1/2 Gamma(n+1). Look up the gamma function and its relation to factorial and you'll find that you basically have the answer there. http://en.wikipedia.org/wiki/Gamma_function
 
  • #5
A_B
93
0
I used a different method ;integrating x*exp(-ax^2), then differentiation wr to a several times. You'll quickely find the pattern. Then you can prove by induction. I get a result that's not in your options. (but it's late, so I could be wrong)

A_B
 
  • #6
3,812
92
Yes, that result is correct. That's good as that makes it equal 1/2 Gamma(n+1). Look up the gamma function and its relation to factorial and you'll find that you basically have the answer there. http://en.wikipedia.org/wiki/Gamma_function
Thanks a lot uart! :smile:
 

Related Threads for: Help for integration

  • Last Post
Replies
2
Views
923
  • Last Post
Replies
7
Views
717
  • Last Post
Replies
1
Views
910
  • Last Post
Replies
5
Views
458
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
3
Views
918
  • Last Post
Replies
15
Views
2K
Replies
1
Views
2K
Top