September 4th, 2011, 12:55 PM  #1 
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  Integral Computation! Compute $\displaystyle \int_{0}^{+\infty}\frac{x^{a1}}{x+1}\;dx$ for $\displaystyle 0<\Re \mathfrak{e}(a)<1$. Last edited by skipjack; February 18th, 2018 at 12:55 AM. 
September 8th, 2011, 04:21 AM  #2 
Newbie Joined: Sep 2011 Posts: 7 Thanks: 0  Re: Integral Computation!
I'm not entirely sure I'm right, but I would start with taking the numerator as u, and denominator as dv. After performing I.B.P you end up with x^(a1)ln abs(x+1) int(ln abs(x+1)*(a1)x^(a2)). Now take (a1)x^(a2) as dv and perform I.B.P again. You should then see that you again get the original unknown integral on RHS. By moving it over you can see that you have 0=0, so your unknown integral should equal [x^(a1)ln abs(x+1)] from 0 to infinity. That integral diverges. 
September 8th, 2011, 04:36 AM  #3 
Newbie Joined: Sep 2011 Posts: 7 Thanks: 0  Re: Integral Computation!
Sorry, it converges.

September 8th, 2011, 04:49 AM  #4 
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: Integral Computation! The result you should come up with is $\displaystyle \pi \cdot\csc(\pi \cdot a )$. Learn how to use $\displaystyle \rm L^{a}T_{e}X$ in order for your posts to be more readable! Last edited by skipjack; February 18th, 2018 at 12:54 AM. 

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