Let a&lt;1 be a positive constant. How can I compute the following integral? ∫ 0 2 π 1 1 + a cos ⁡ ( x ) d x

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Let a&amp;lt;1 be a positive constant. How can I compute the following integral?      ∫          0              2      π            1          1      +      a                  cos      ⁡      (      x      )        d  x

xxsailojaixxv5 · Accepted Answer

For       |    A      |    ≥  1, the integral diverges. For       |    A      |    &amp;lt;  1, we can use the Weierstrass substitution (or alternatively, use contour integration) to evaluate the integral.Proceeding, we first write      ∫    0          2      π            1          1      +      A      cos      ⁡      (      x      )          d  x  =  2      ∫    0          π            1          1      +      A      cos      ⁡      (      x      )          d  xThen, letting   t  =  tan  ⁡  (  x      /    2  ) so that   cos  ⁡  (  x  )  =            1      −              t        2                    1      +              t        2             and   d  x  =      2          1      +              t        2                d  t, we obtain  2      ∫    0          π            1          1      +      A      cos      ⁡      (      x      )          d  x  =  4      ∫    0    ∞        1          (      1      −      A      )              t        2            +      1      +      A          d  t

Let a<1 be a positive constant. How can I compute the following integral? <msubsup> &#x222

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