I have to integrate this : ∫ cos 2 ⁡ ( x ) 1 + tan ⁡ ( x ) d x

Question

I have to integrate this :   ∫                    cos        2            ⁡      (      x      )              1      +      tan      ⁡      (      x      )        d  x

Elliott Gilmore · Accepted Answer

We need to calculate  I  =  ∫                    cos        2            ⁡      x              1      +      tan      ⁡      x          d  x  =  ∫                    cos        3            ⁡      x              sin      ⁡      x      +      cos      ⁡      x          d  x  .Consider another integral  J  =  ∫                    sin        3            ⁡      x              sin      ⁡      x      +      cos      ⁡      x          d  xand calculate I+J and I-J using      a    3    ±      b    3    =  (  a  ±  b  )  (      a    2    ∓  a  b  +      b    2    )  .                              I          +          J                                    =        ∫        (                  cos          2                ⁡        x        −        cos        ⁡        x        sin        ⁡        x        +                  sin          2                ⁡        x        )                d        x        =        ∫                  (          1          −                      1            2                    sin          ⁡          2          x          )                        d        x        =                  x          +                                    cos              ⁡              2              x                        4                          +        C        ,                                      I          −          J                                    =        ∫                              cos            ⁡            x            −            sin            ⁡            x                                sin            ⁡            x            +            cos            ⁡            x                          ⋅        (                  cos          2                ⁡        x        +        cos        ⁡        x        sin        ⁡        x        +                  sin          2                ⁡        x        )                d        x        =                                          =        ∫                              cos            ⁡            x            −            sin            ⁡            x                                sin            ⁡            x            +            cos            ⁡            x                          ⋅                              (            cos            ⁡            x            +            sin            ⁡            x                          )              2                        +            1                    2                        d        x        =                                          =                  1          2                ∫        cos        ⁡        2        x                d        x        +                  1          2                ∫                              cos            ⁡            x            −            sin            ⁡            x                                sin            ⁡            x            +            cos            ⁡            x                                  d        x        =                                            sin              ⁡              2              x                        4                    +                      1            2                    ln          ⁡                      |                    sin          ⁡          x          +          cos          ⁡          x                      |                          +        C        .            Now add       I    +    J   and       I    −    J   and divide by two.

I have to integrate this : ∫ cos 2 <

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