Compute I n = lim n → ∞ n ∫ 0 π 2 sin n ⁡ x d x

Question

Compute       I    n    =      lim          n      →      ∞            n        ∫          0                      π        2                  sin          n        ⁡  x  d  x

trajeronls · Accepted Answer

You may show that the main contribution comes from the neighbourhood of       π    2  . Then, by dominated convergence,                              n                          ∫                      π                          /                        2            −            ε                                π                          /                        2                                                sin            n                    ⁡          x                    d          x                                    ≈                  n                          ∫                      π                          /                        2            −            ε                                π                          /                        2                                                              (                              1                −                                  1                  2                                                                      (                                                                  π                        2                                            −                      x                                        )                                    2                                            )                        n                    d          x                =                  n                          ∫          0          ε                                                    (                              1                −                                  1                  2                                                  t                  2                                            )                        n                    d          t                                                  =                  ∫          0                                    n                        ε                                                              (                              1                −                                                                            s                      2                                                                            2                    n                                                              )                        n                    d          s                →                  ∫          0                      +            ∞                                                e                          −                              s                2                                            /                            2                                d          s                =                              π            2                          .            I let you make this argument rigorous. You may study the Laplace method.

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New Questions in Calculus 1