How to prove that ∫ 0 1 ( 1 + x n ) − 1 − 1 / n d x = 2 − 1 / n

Question

How to prove that      ∫          0              1        (  1  +      x    n        )          −      1      −      1              /            n        d  x  =      2          −      1              /            n

Zackary Hatfield · Accepted Answer

The change of variable   t  =      x          −      n       yields       d    t  =  −  n      x          −      n      −      1            d    x, that is,       d    x  =  −      1    n        t          −      1      −      1              /            n            d    t. Thus, the integral is      ∫    1          +      ∞            1    n                      d            t              (      1      +      t              )                  1          +          1                      /                    n                      =            [                        −          1                          (          1          +          t                      )                          1                              /                            n                                          ]        1          +      ∞        =      1          2              1                  /                n              .

evitagimm9h · Accepted Answer

Let   x  =  (  tan  ⁡      t        )          2              /            n      . Then the integral becomes                              ∫          0          1                d        x                (        1        +                  x          n                          )                      −                          (              1              +                              1                n                            )                                                          =                  2          n                          ∫          0                      π                          /                        4                          d        t                cot        ⁡                  t                (        tan        ⁡                  t                          )                      2                          /                        n                          (        sec        ⁡                  t                          )                      −            2                          /                        n                                                            =                  2          n                          ∫          0                      π                          /                        4                          d        t                cot        ⁡                  t                (        sin        ⁡                  t                          )                      2                          /                        n                                                            =                  2          n                          ∫          0                      π                          /                        4                          d        (        sin        ⁡                  t                )                (        sin        ⁡                  t                          )                      2                          /                        n            −            1                                                            =        [        (        sin        ⁡                  t                          )                      2                          /                        n                                    ]          0                      π                          /                        4                                                            =                  2                      −            1                          /                        n

How to prove that int_0^1(1+x^n)^(-1-1/n)dx=2^(-1/n)

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