November 21st, 2015, 09:45 PM  #1 
Newbie Joined: Nov 2015 From: iran Posts: 1 Thanks: 0  solve integral
hello could you solve this integral using complex function infinite integral(1/(x^3+1)^2)dx (0 to unlimited) THANKS 
November 23rd, 2015, 05:27 PM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,492 Thanks: 632 
That looks like a pretty standard kind of complex analysis problem. Set up a contour integral in which the contour goes from 0 to R along the positive real axis, along a quarter circle with center at 0 and radius R from R to Ri, then from Ri to 0 along the positive complex axis. Along the real axis, taking z= x, it is the desired [tex]\int_0^R \frac{1}{(x^3+ 1)}dx. Along the complex axis, taking z= xi, it is . The integral along the quarter circle is harder to set up but clearly goes to 0 as R goes to 0. The function is analytic inside the contour so the entire integral, the sum of those three, is 0.


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