Given an integral like the following: ∫ 0 1 x n ( 1 − x ) n d x and the substitution like x = sin 2 ⁡ θWhat would the limits be?

Question

Given an integral like the following:      ∫    0    1        x    n    (  1  −  x      )    n    d  x and the substitution like   x  =      sin    2    ⁡  θWhat would the limits be?

tabustudiofx52n · Accepted Answer

By Euler&#039;s beta function,                    (1)                              ∫                      0                                1                                    x          n                (        1        −        x                  )          n                        d        x        =                              Γ            (            n            +            1                          )              2                                            Γ            (            2            n            +            2            )                          =                  1                      (            2            n            +            1            )                                                            (                                                                              2                  n                                n                                                              )                                                                    .            (1) may be checked by integration by parts, no need of trigonometric substitutions.

luyethabakcync · Accepted Answer

By the mean-value theorem we have  0  ≤      ∫          x      =      0              1            x          n        (  1  −  x      )          n        ≤      sup          t      ∈      ]      0      ,      1      [            t          n        (  1  −  t      )          n        →  0.

Given an integral like the following: <msubsup> ∫ 0 1 </ms

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