Integrate using integration by parts ∫ e − x 2 1 − sin ⁡ x 1 + cos ⁡ x d x

Question

Integrate using integration by parts   ∫      e          −              x        2                        1      −      sin      ⁡              x                    1      +      cos      ⁡              x              d  x

knolsaadme · Accepted Answer

We first consider the half-angle transform            1      −      sin      ⁡              x                    1      +      cos      ⁡              x              =            cos      ⁡              x        2            −      sin      ⁡              x        2                    2              cos        2            ⁡              x        2            Notice that            d                      d            x            1          cos      ⁡              x        2              =            sin      ⁡              x        2                    2              cos        2            ⁡              x        2            So we have                                  ∫                  e                      −                          x              2                                                            1            −            sin            ⁡                          x                                            1            +            cos            ⁡                          x                                                d                x        =        ∫                  e                      −                          x              2                                                            cos            ⁡                          x              2                        −            sin            ⁡                          x              2                                            2                          cos              2                        ⁡                          x              2                                                d                x                            =                            ∫                  e                      −                          x              2                                                            cos            ⁡                          x              2                                            2                          cos              2                        ⁡                          x              2                                                d                x        −        ∫                  e                      −                          x              2                                                            sin            ⁡                          x              2                                            2                          cos              2                        ⁡                          x              2                                                d                x                            =                            ∫                  e                      −                          x              2                                                1                      2            cos            ⁡                          x              2                                                d                x        −                  e                      −                          x              2                                                1                      cos            ⁡                          x              2                                      −        ∫                  1          2                          e                      −                          x              2                                                1                      cos            ⁡                          x              2                                                d                x        =        −                  e                      −                          x              2                                                1                      cos            ⁡                          x              2

Integrate using integration by parts ∫ e <mrow class="MJX-TeXAtom-ORD

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