Evaluate ∫ sin 3 ⁡ ( θ / 2 ) cos ⁡ ( θ / 2 ) cos 3 ⁡ θ + cos 2 ⁡ θ + cos ⁡ θ d θ

Question

Evaluate   ∫                    sin        3            ⁡      (      θ              /            2      )              cos      ⁡      (      θ              /            2      )                        cos          3                ⁡        θ        +                  cos          2                ⁡        θ        +        cos        ⁡        θ              d  θ

hofyonlines5 · Accepted Answer

Let      I    =    ∫                            sin          3                ⁡        (        θ                  /                2        )                    cos        ⁡        (        θ                  /                2        )                              cos            3                    ⁡          θ          +                      cos            2                    ⁡          θ          +          cos          ⁡          θ                      d    θ    =          1      2        ∫                  2                  sin          2                ⁡                  θ          2                ⋅        2        sin        ⁡                  θ          2                ⋅        cos        ⁡                  θ          2                            2                  cos          2                ⁡                  θ          2                                      cos            3                    ⁡          θ          +                      cos            2                    ⁡          θ          +          cos          ⁡          θ                      d    θ  So we get      I    =          1      2        ∫                  (        1        −        cos        ⁡        θ        )        ⋅        sin        ⁡        θ                    (        1        +        cos        ⁡        θ        )                              cos            3                    ⁡          θ          +                      cos            2                    ⁡          θ          +          cos          ⁡          θ                      d    θ  Now Put   cos  ⁡  θ  =  t    ,, Then   sin  ⁡  θ  d  θ  =  −  d  tSo Integral      I    =    −          1      2        ∫                  (        1        −        t        )                    (        1        +        t        )                              t            3                    +                      t            2                    +          t                      d    t    =    −          1      2        ∫                  (        1        −                  t          2                )                    (        1        +        t                  )          2                                      t            3                    +                      t            2                    +          t                      d    t  So we get      I    =          1      2        ∫                  (        1        −                  1                      t            2                          )                              (          t          +                      1            t                    +          2          )                          t          +                      1            t                    +          1                      d    t  Now Let             (      t      +              1        t            +      1      )        =          u      2            ,  , Then       (    1    −          1              t        2              )    d  t  =  2  u  d  uSo Integral      I    =          1      2        ∫                  2        u                              u          2                +        1              ⋅          1      u        d    u    =          tan              −        1              ⁡    (    u    )    +          C      So we get      I    =          tan              −        1              ⁡                  (        t        +                  1          t                +        1        )              +          C      So we get            I      =              tan                  −          1                    ⁡                        (          cos          ⁡          θ          +          sec          ⁡          θ          +          1          )                    +              C

Evaluate ∫ sin 3 </msup> &#x2

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