Integrate sin ⁡ ( 3 x ) cos ⁡ ( 3 x )

Question

Integrate   sin  ⁡  (  3  x  )  cos  ⁡  (  3  x  )

poquetahr · Accepted Answer

∫  sin  ⁡  3  x  cos  ⁡  3  x            d        x    =  ∫      1    2    sin  ⁡  6  x            d        x    =  −      1    12    cos  ⁡  6  x  +  C

kramberol · Accepted Answer

Here&#039;s one way to do this :                    ∫        sin        ⁡        (        3        x        )        cos        ⁡        (        3        x        )                              d                    x                                    =                  1          3                ∫        3        sin        ⁡        (        3        x        )        cos        ⁡        (        3        x        )                              d                    x                                                  =                  1          3                ∫        sin        ⁡        (        3        x        )        (        sin        ⁡        (        3        x        )                  )          ′                                      d                    x                                                  =                  1          6                ∫        (        sin        ⁡        (        3        x                  )          2                          )          ′                                      d                    x                                                  =                  1          6                sin        ⁡        (        3        x                  )          2                +        K            This is in fact the same as doing a substitution   u  =  sin  ⁡  (  3  x  ).Note also that though it may seem different than almagest&#039;s solution, the two are equivalent. Indeed,  cos  ⁡  (  6  x      )    2    =  1  −  2  sin  ⁡  (  3  x      )    2    ,so that  −      1    12    cos  ⁡  (  6  x  )  +  C  =  −      1    12    (  1  −  2  sin  ⁡  (  3  x      )    2    )  +  C  =      1    6    sin  ⁡  (  3  x      )    2    +  C  −      1    12    ,which is the same as my answer if you choose   K  =  C  −      1    12

Integrate sin ⁡ (

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