How to evaluate ∫ x n e x cos ⁡ x d x and ∫ x n e x sin ⁡ x d x ?

Question

How to evaluate   ∫      x    n        e          x        cos  ⁡  x  d  x and   ∫      x    n        e          x        sin  ⁡  x  d  x  ?

Jarrett Reyes · Accepted Answer

Note that                    ∫                  x          n                          e                      a            x                          d        x        =                              d            n                                d                          a              n                                      ∫                  e                      a            x                          d        x        =                              d            n                                d                          a              n                                                (                                    e                              a                x                                      a                    )                =                  ∑                      k            =            0                    n                                      (            −            1                          )              k                        n            !                                (            n            −            k            )            !                                                              x                              n                −                k                                                    e                              a                x                                                          a                          k              +              1                                          Then                    ∫                  x          n                          e                      x                          cos        ⁡        x                d        x        =                            ℜ        ∫                  x          n                          e                      (            1            +            i            )            x                          d        x        =                ℜ                  ∑                      k            =            0                    n                                      (            −            1                          )              k                        n            !                                (            n            −            k            )            !                                                              x                              n                −                k                                                    e                              (                1                +                i                )                x                                                          (            1            +            i                          )                              k                +                1                                                                        =                                      ∑                      k            =            0                    n                                      (            −            1                          )              k                        n            !                          x                              n                −                k                                                    e                              x                                                                        2                                                                    k                    +                    1                                    2                                                      (            n            −            k            )            !                          cos        ⁡                              (                          x        −                              (            k            +            1            )            π                    4                                      )                                                            ∫                  x          n                          e                      x                          sin        ⁡        x                d        x        =                                      ∑                      k            =            0                    n                                      (            −            1                          )              k                        n            !                          x                              n                −                k                                                    e                              x                                                                        2                                                                    k                    +                    1                                    2                                                      (            n            −            k            )            !                          sin        ⁡                              (                          x        −                              (            k            +            1            )            π                    4                                      )

How to evaluate ∫ x n </msup> e <mrow class="MJX-TeXAtom

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