I am trying to compute this integral ∫ 0 ∞ x 1 + x 6 d x using the residue theorem. To do so, I am integrating f ( z ) = z 1 + z 6 in the frontier of this sector of a circle: { z : | z | &lt; R , 0 &lt; a r g ( z ) &lt; π / 3 }.I kwo how to deal with the integrals over the horizontal segment of the sector and over the arc of the circle. My problem is the "diagonal segment". When I parametrize it, I do not get something easy to integrate. How could I approach this?(The path of integration was suggested in my book, so I do not think that is the problem).

Question

I am trying to compute this integral       ∫          0              ∞            x          1      +              x        6              d  x using the residue theorem. To do so, I am integrating   f  (  z  )  =      z          1      +              z        6             in the frontier of this sector of a circle:   {  z  :      |    z      |    &amp;lt;  R  ,  0  &amp;lt;  a  r  g  (  z  )  &amp;lt;  π      /    3  }.I kwo how to deal with the integrals over the horizontal segment of the sector and over the arc of the circle. My problem is the &quot;diagonal segment&quot;. When I parametrize it, I do not get something easy to integrate. How could I approach this?(The path of integration was suggested in my book, so I do not think that is the problem).

Jaxson White · Accepted Answer

The paramertic form is   z  =  r  (            1      +              3              2    )  ,  0  &amp;lt;  r  &amp;lt;  R. The integral on this line becomes   (            1      +              3              2    )      ∫    0          R            r          1      +              r                  6                      d  r since   (            1      +              3              2        )          6        =  1. Now you get   [  1  −            1      +              3              2    )  ]      ∫    0          ∞            r          1      +              r                  6                      d  r  = times the sum of the residues.

I am trying to compute this integral int^infty_0 (x)/(1+x^6)dx using the residue theorem.

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