Solve ∫ x sin 2 ⁡ ( x ) d x

Question

Solve   ∫      x          sin      2        ⁡    (    x    )       d  x

Pettanicej4lyy · Accepted Answer

Hint:- Let  I  =      ∫    x          sin      2        ⁡    x         d    x    J  =      ∫    x          cos      2        ⁡    x         d    x  Now observe that,  I  +  J  =      ∫    x         d    x  and  I  −  J  =  ℜ      (          ∫      x              e                  2          i          x                           d      x        )  where   ℜ      (          ∫      x              e                  2          i          x                           d      x        )   denotes the real part of the integral

Ayana Waller · Accepted Answer

∫      x          sin              2              ⁡    (    x    )    d  x  =  ∫            sin              2              ⁡    (    x    )    d          (                                    x                                2                          2            )        =                                        x                                2                                    sin                      2                          ⁡        (        x        )            2        −      1    2    ∫                    x                    2              sin    ⁡          2      x        d    x    =    =                              x                          2                            sin                  2                    ⁡      (      x      )        2    +      1    4    ∫                    x                    2              d    cos    ⁡          2      x        =                                        x                                2                                    sin                      2                          ⁡        (        x        )            2        +          1      4              (                        x                          2                    cos      ⁡              2        x            −      2      ∫              x        cos        ⁡                  2          x          d          x                    )        =      =                              x                          2                            sin                  2                    ⁡      (      x      )        2    +      1    4        (                  x                    2              cos    ⁡          2      x        −    ∫          x      d      sin      ⁡              2        x              )    =                              x                          2                            sin                  2                    ⁡      (      x      )        2    +      1    4        (                  x                    2              cos    ⁡          2      x        −    x    sin    ⁡          2      x      +      ∫              sin        ⁡                  2          x          d          x                      )    =    =                              x                          2                            sin                  2                    ⁡      (      x      )        2    +                              x                          2                    cos      ⁡              2        x        −            x      sin      ⁡              2        x              4    −      1    8    cos  ⁡      2    x    +  C

Solve ∫ <mrow class="MJX-TeXAtom-ORD"> x sin 2

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