I've F ( t ) = ∫ 0 ∞ e − t x 3 1 + x 4 d x and I have to see that it is well defined on the interval ( 0 , ∞ ).For that, I have defined f ( x , t ) = e − t x 3 1 + x 4 , x , t ∈ ( 0 , ∞ ) so I have to see if f is integrable in ( 0 , ∞ ).We know that f is integrable on ∫ ( 0 , ∞ ) | f | d μ &lt; ∞ f ( x , t ) = | e − t x 3 1 + x 4 | = e − t x 3 1 + x 4 ≤ e − t x 3 But hoe can I bound this? I have to bound it with an integrable function... but I don't know how to calculate the integral of e − t x 3 ... Is there any other easier way to bound that? Or how can I solve my problem?

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I&#039;ve   F  (  t  )  =      ∫    0          ∞                  e              −        t                  x          3                            1      +              x        4              d  x and I have to see that it is well defined on the interval   (  0  ,  ∞  ).For that, I have defined   f  (  x  ,  t  )  =            e              −        t                  x          3                            1      +              x        4              ,  x  ,  t  ∈  (  0  ,  ∞  ) so I have to see if   f is integrable in   (  0  ,  ∞  ).We know that   f is integrable on       ∫          (      0      ,      ∞      )            |    f      |    d  μ  &amp;lt;  ∞  f  (  x  ,  t  )  =      |              e              −        t                  x          3                            1      +              x        4                  |    =            e              −        t                  x          3                            1      +              x        4              ≤      e          −      t              x        3            But hoe can I bound this? I have to bound it with an integrable function... but I don&#039;t know how to calculate the integral of       e          −      t              x        3            ... Is there any other easier way to bound that? Or how can I solve my problem?

Immanuel Glenn · Accepted Answer

Hint:       ∫    0          1            e          −      t              x                  3                      d  x  ≤      ∫    0          1        1  d  x and       ∫    1          ∞            e          −      t              x                  3                      d  x  ≤      ∫    1          ∞            e          −      t      x        d  xTo prove continuity of   F it is enough to prove continuity on (r,∞) for each r&amp;gt;0. When   t  &amp;gt;  r we have       |    f  (  x  ,  t  )      |    ≤  max  {      1          0      &amp;lt;      x      &amp;lt;      1        ,      e          −      r      x        }. Now apply DCT to prove sequential continuity of   F.

I've F ( t ) = <msubsup> ∫ 0 <mrow class="MJX-TeX

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