Integrate ∫ 1 1 + sin ⁡ x d x using substitution u = 1 + sin ⁡ x

Question

Integrate    ∫      1          1      +      sin      ⁡      x          d  x using substitution   u  =  1  +  sin  ⁡  x

Cordell Crosby · Accepted Answer

Note that   d  u  =  cos  ⁡  x    d  x  =  sgn  (  cos  ⁡  x  )      2    u    −          u      2        d  x and your solution is                    ∫                              1                          1              +              sin              ⁡              x                                              d        x                            =        sgn        (        cos        ⁡        x        )        ∫                              1            u                          ×                              1                          2              u              −                              u                2                                                            d        u                                          =        −        sgn        (        cos        ⁡        x        )                                            2                              1                +                sin                ⁡                x                                              −          1                +        C                                          =        −        sgn        (        cos        ⁡        x        )                                                            1                −                sin                ⁡                x                                            1                +                sin                ⁡                x                                                    +        C                                          =        −        sgn        (        cos        ⁡        x        )                                                                              |                                cos                ⁡                x                                  |                                                            1                +                sin                ⁡                x                                                    +        C                                          =        −                                                            cos                ⁡                x                                            1                +                sin                ⁡                x                                                    +        C

Integrate ∫ 1 1 + sin ⁡

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