December 7th, 2011, 01:34 PM  #11  
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Nine Integrals Quote:
 
December 29th, 2011, 06:38 PM  #12 
Senior Member Joined: Dec 2011 From: Argentina Posts: 216 Thanks: 0  Re: Nine Integrals
I'd like to contribute with 5. Now by integrating by parts with we get that Thus we get that for even for odd Now So writing as or we get that. which you can always write in terms of the Gamma function and factorials knowing that 
December 29th, 2011, 07:29 PM  #13 
Senior Member Joined: Dec 2011 From: Argentina Posts: 216 Thanks: 0  Re: Nine Integrals
Why is the 3rd integral non zero if the function being integrated is even? My doubt arises since for any even function: 
December 29th, 2011, 07:30 PM  #14 
Senior Member Joined: Dec 2011 From: Argentina Posts: 216 Thanks: 0  Re: Nine Integrals
I'm sorry, I meant for odd functions, but the function in question is even, so never mind.

December 29th, 2011, 08:00 PM  #15 
Senior Member Joined: Dec 2011 From: Argentina Posts: 216 Thanks: 0  Re: Nine Integrals
I'd like to provide a proof for Theorem Proof Substitute to get Differentiating under Leibniz's Rule: Given that and that we have got but again, by Lebniz's Rule: so satisfies the differential equation thus and since we have QED Although this proof is not really rigorous, I like it. 
December 29th, 2011, 08:03 PM  #16 
Senior Member Joined: Dec 2011 From: Argentina Posts: 216 Thanks: 0  Re: Nine Integrals
And 2 follows from 3 since 2 is the negative integral of 3.

December 30th, 2011, 06:33 PM  #17 
Senior Member Joined: Dec 2011 From: Argentina Posts: 216 Thanks: 0  Re: Nine Integrals
Ok here goes another one: Now Thus Then 
January 2nd, 2012, 01:00 PM  #18 
Global Moderator Joined: Dec 2006 Posts: 21,036 Thanks: 2274 
More directly, ? if I = ? ln(x)/(x² + a²) dx, where a > 0, 0 let x = a²/u, dx = a²/u² du. ? ? ? I = ? ln(a²/u)/((a²/u)² + a²) (a²/u²) du = ? ln(a²/u)/(a² + u²) du = ? ln(a²)/(a² + u²) du  I. 0 0 0 ? Hence I = (1/2)ln(a²)[(1/a)atan(u/a)] = (1/a)ln(a)?/2. 0 
January 2nd, 2012, 05:46 PM  #19 
Senior Member Joined: Dec 2011 From: Argentina Posts: 216 Thanks: 0  Re: Nine Integrals
Nice one. I'm trying to polish my proof for 3.


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