How to integrate ∫ 1 cos ⁡ ( x ) d x using the substitution u = tan ⁡ ( x 2 ) ?

Question

How to integrate   ∫            1              cos        ⁡        (        x        )              d  x using the substitution   u  =  tan  ⁡      (          x      2        )  ?

redclick53 · Accepted Answer

Bioche&#039;s rules say in this case the correct substitution is   u  =  sin  ⁡  x,       d      u  =  cos  ⁡  x        d      x. Indeed  ∫                    d                  x              cos      ⁡      x        =  ∫            cos      ⁡      x                    d                  x                      cos        2            ⁡      x        =  ∫                    d                  u              1      −              u        2              =      1    2    ln  ⁡            (                  1      +      u              1      −      u                  )        =      1    2    ln  ⁡            (                  1      +      sin      ⁡      x              1      −      sin      ⁡      x                  )        .Note:Using some trigonometry formulae, this may be rewritten as  ln  ⁡            (        tan  ⁡            (            x    2    +      π    4              )                  )        .

Ashly Harrell · Accepted Answer

Hint:If   u  =  tan  ⁡      (          x      2        )  , then   cos  ⁡  x  =                    1        −                  u          2                            1        +                  u          2                     and   d  x  =                    2                 d        u                    1        +                  u          2                    . Hence  ∫            1              cos        ⁡        x                 d  x  =  ∫            2              1        −                  u          2                         d  u  =  ∫      [          1              1        +        u              +          1              1        −        u              ]       d  u

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