Evaluate ∫ 0 π log ⁡ ( 5 − 4 cos ⁡ x ) d x

Question

Evaluate       ∫    0          π        log  ⁡      (    5    −    4    cos    ⁡    x    )          d    x

Patrick Arnold · Accepted Answer

Let   I  (  a  )  =      ∫    0          π        log  ⁡      (    1    −    a    cos    ⁡    x    )          d    x. Then      I    ′    (  a  )  =  −      ∫    0    π              cos      ⁡      x              1      −      a      cos      ⁡      x        d  x  =      π    a    −      π          a              1        −                  a          2                    and                              ∫          0                      π                          log        ⁡                  (          5          −          4          cos          ⁡          x          )                                  d                x        =                            π        ln        ⁡        5        +        I                              (                                    4          5                                      )                          =        π        ln        ⁡        5        +                  ∫          0                                    4              5                                                I          ′                (        a        )        d        a                            =                            π        ln        ⁡        5        −        π        ln        ⁡                  5          4                =        2        π        ln        ⁡        2

Evaluate int_0^(pi) log(5-4 cos x)dx

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Answer & Explanation

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