When ∫ − π π | x | α | tan ⁡ x | β d x &lt; ∞ and ∫ − π π | tan ⁡ x | γ d x &lt; ∞? Closed forms?For which α &gt; 0 , β ∈ ( 0 , 2 ) , γ ∈ ( 0 , 2 ) do we have ∫ − π π | x | α | tan ⁡ x | β d x &lt; ∞ ∫ − π π | tan ⁡ x | γ d x &lt; ∞ ?Can you also give a closed-form expression for the antiderivatives (or the definite integrals)?

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tabustudiofx52n · Accepted Answer

Step 1The problem occurs at   x  =  ±  π      /    2. As   x  →  π      /    2 we have   tan  ⁡  (  x  )  ∼  −  (  x  −  π      /    2      )          −      1      , so your integrals will be finite if   β  &amp;lt;  1 and   γ  &amp;lt;  1.Step 2I would not expect there to be closed-form antiderivatives in the first integral. For the second, I get  ∫  tan  ⁡  (  x      )    γ      d  x  =            tan      ⁡      (      x              )                  γ          +          1                      2    Φ      (    −                  tan        ⁡        (        x        )                    2              ,    1    ,                  γ        +        1            2        )  where   Φ is the Lerch Phi function, defined by  Φ  (  z  ,  a  ,  v  )  =      ∑          k      =      0              ∞                  z      k              (      v      +      k              )        a            For rational   γ you&#039;ll get more elementary forms of this antiderivative.

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