I'm having some trouble figuring out the right substitutions to make to integrate ∫ sin 4 ⁡ ( θ ) d θand ∫ cos 4 ⁡ ( θ ) d θAny hints or suggestions are welcome.

Question

I&#039;m having some trouble figuring out the right substitutions to make to integrate  ∫      sin    4    ⁡  (  θ  )  d  θand  ∫      cos    4    ⁡  (  θ  )  d  θAny hints or suggestions are welcome.

Quinn Everett · Accepted Answer

Notice that you can write       sin    4    ⁡  (  x  ) as follows:                              sin          4                ⁡        (        x        )        =                             (                  sin          2                ⁡        (        x        )                  )          2                =                              (                                                1            −            cos            ⁡            (            2            x            )                    2                                                    )                                2                                    =                                                   1            −            2            cos            ⁡            (            2            x            )            +                          cos              2                        ⁡            (            2            x            )                    4                                    =                                       1          4                −                              cos            ⁡            (            2            x            )                    2                +                              1            +            cos            ⁡            (            4            x            )                    8                .            For       cos    4    ⁡  (  x  ) we procede as above:                              cos          4                ⁡        (        x        )        =                             (                  cos          2                ⁡        (        x        )                  )          2                =                              (                                                1            +            cos            ⁡            (            2            x            )                    2                                                    )                                2                                    =                                                   1            +            2            cos            ⁡            (            2            x            )            +                          cos              2                        ⁡            (            2            x            )                    4                                    =                                       1          4                +                              cos            ⁡            (            2            x            )                    2                +                              1            +            cos            ⁡            (            4            x            )                    8                .

deceptie3j · Accepted Answer

HINT (using partial integration):  ∫      sin    4    ⁡  (  x  )        d  x  =  −      1    4        sin    3    ⁡  (  x  )  cos  ⁡  (  x  )  +      3    4    ∫      sin    2    ⁡  (  x  )        d  x  =  −      1    4        sin    3    ⁡  (  x  )  cos  ⁡  (  x  )  +      3    4    ∫      (          1      2        −          1      2        cos    ⁡    (    2    x    )    )          d  x  =  −      1    4        sin    3    ⁡  (  x  )  cos  ⁡  (  x  )  −      3    8    ∫  cos  ⁡  (  2  x  )        d  x  +      3    8    ∫  1        d  x  ∫      cos    4    ⁡  (  x  )        d  x  =      1    4    sin  ⁡  (  x  )      cos    3    ⁡  (  x  )  +      3    4    ∫      cos    2    ⁡  (  x  )        d  x  =      1    4    sin  ⁡  (  x  )      cos    3    ⁡  (  x  )  +      3    4    ∫      (          1      2        +          1      2        cos    ⁡    (    2    x    )    )          d  x  =      1    4    sin  ⁡  (  x  )      cos    3    ⁡  (  x  )  +      3    8    ∫  cos  ⁡  (  2  x  )        d  x  +      3    8    ∫  1        d  x

I'm having some trouble figuring out the right substitutions to make to integrate ∫<!-- ∫

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