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November 21st, 2015, 08:45 PM   #1
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From: iran

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solve integral

could you solve this integral using complex function

infinite integral(1/(x^3+1)^2)dx (0 to unlimited)

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November 23rd, 2015, 04:27 PM   #2
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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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