Determine this real integral with the Residue-theorem. ∫ − ∞ ∞ sin ⁡ x x 4 − 6 x 2 + 10 d x

Question

Determine this real integral with the Residue-theorem.      ∫          −      ∞              ∞                          sin        ⁡        x                              x          4                −        6                  x          2                +        10                        d        x

enfujahl · Accepted Answer

Hint. Since       x    4    −  6      x    2    +  10 has no real zeros try to replace   sin  ⁡  x with             e                      i            x       via the Euler formula and take only the imaginary part of it like      ∫          −      ∞        ∞              sin      ⁡      x                      x        4            −      6              x        2            +      10              d    x  =  Im  ⁡      (          ∫              −        ∞            ∞                                e                                      i                    x                                      x          4                −        6                  x          2                +        10                        d        x    )  since you will get a Fourier integral of the form      ∫          −      ∞        ∞              P      (      z      )              Q      (      z      )                  e                      i            z              d    z  =  2  π      i        ∑          z      ∈              Q                  −          1                    (      {      0      }      )      ,      Im      ⁡      z      &amp;gt;      0            Res          w      =      z        ⁡      (                  P        (        w        )                    Q        (        w        )                            e                              i                w              )    .However the given integral has value zero which can be shown in multiple ways.

Determine this real integral with the Residue-theorem. <msubsup> ∫ <

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