Integral inequality: 1 30 &lt; ∫ 2 ∞ s 3 − s 2 + 3 s 5 + s 2 + 1 d s &lt; 2 20

Question

Integral inequality:       1    30    &amp;lt;      ∫          2              ∞                          s        3            −              s        2            +      3                      s        5            +              s        2            +      1          d  s  &amp;lt;            2        20

mignonechatte00f · Accepted Answer

For right inequality, write      ∫          2              ∞                                    s          3                −                  s          2                +        3                              s          5                +                  s          2                +        1                  d    s  &amp;lt;      ∫          2              ∞                                    s          3                            s        5                  d    s  =      ∫          2              ∞            s          −              7        2                  d    s  =            2        20  For left one, you may try      ∫          2              ∞                                    s          3                −                  s          2                +        3                              s          5                +                  s          2                +        1                  d    s  &amp;gt;      ∫          2              ∞                          7                              s          5                +                  s          3                +                  s          2                +        1                  d    sThen using the fact that       s    5    +      s    3    +      s    2    +  1  =  (  s  +  1  )  (      s    2    +  1  )  (      s    2    −  s  +  1  ), you have      1                  s        5            +              s        3            +              s        2            +      1        =  −            2      s      −      1              3      (              s        2            −      s      +      1      )        +            s      +      1              2      (              s        2            +      1      )        +      1          6      (      s      +      1      )      Then you can compute the integral,      ∫          2              ∞                          7                              s          5                +                  s          3                +                  s          2                +        1                  d    s  =      7        (    −                  log        ⁡        5            4        +                  log        ⁡        3            6        +                  1        2              arctan    ⁡                  1        2              )

Prove that 1/30<int_2^infty (sqrt(s^3-s^2+3))/(s^5+s^2+1)ds< sqrt2/20

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