Tips for integrating ∫ d x 1 + cos ⁡ ( x )

Question

Tips for integrating   ∫            d      x              1      +      cos      ⁡      (      x      )

upornompe · Accepted Answer

Do the substitution x=2u.  ∫            d      x              1      +      cos      ⁡      (      x      )        =  2  ∫            d      u              1      +      cos      ⁡      (      2      u      )        =  ∫            d      u                      cos        2            ⁡      (      u      )        =  tan  ⁡  (  u  )  +  C  =  tan  ⁡  (  x      /    2  )  +  C  .

Boilanubjaini8f · Accepted Answer

Recalling   1  =      cos    2    ⁡      x    2    +      sin    2    ⁡      x    2   and   cos  ⁡  x  =      cos    2    ⁡      x    2    −      sin    2    ⁡      x    2  , the integral may be written as                    ∫                              d            x                                1            +            cos            ⁡            x                                              =        ∫                              d            x                                              (                              cos                2                            ⁡                              x                2                            +                              sin                2                            ⁡                              x                2                            )                        +                          (                              cos                2                            ⁡                              x                2                            −                              sin                2                            ⁡                              x                2                            )                                                                        =        ∫                              d            x                                2                          cos              2                        ⁡                          x              2                                                                        =                  1          2                ∫                  sec          2                ⁡                  x          2                        d        x                                          =        tan        ⁡                  x          2                +                  C

Tips for integrating ∫ d x </mrow> 1

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