Show that ∫ 0 π / 2 log 2 ⁡ sin ⁡ x log 2 ⁡ cos ⁡ x cos ⁡ x sin ⁡ x d x = 1 4 ( 2 ζ ( 5 ) − ζ ( 2 ) ζ ( 3 ) )

Question

Show that       ∫          0              π              /            2                          log        2            ⁡      sin      ⁡      x              log        2            ⁡      cos      ⁡      x              cos      ⁡      x      sin      ⁡      x            d    x  =      1    4        (    2    ζ    (    5    )    −    ζ    (    2    )    ζ    (    3    )    )

Samsonitew7b · Accepted Answer

Answer:                    I                            =                  ∫          0                      π                          /                        2                                                              ln              2                        ⁡            cos            ⁡            x                          ln              2                        ⁡            sin            ⁡            x                                cos            ⁡            x            sin            ⁡            x                                   d        x                  =                      sin            ⁡            x            =            u                                    1          4                          ∫          0          1                                                    ln              2                        ⁡            (            1            −                          x              2                        )                          ln              2                        ⁡            x                                x            (            1            −                          x              2                        )                                   d        x                                          =                  1          32                          ∫          0          1                                                    ln              2                        ⁡            (            1            −            x            )                          ln              2                        ⁡            x                                x            (            1            −            x            )                                   d        x                                          =                  1          32                          ∫          0          1                                                    ln              2                        ⁡            (            1            −            x            )                          ln              2                        ⁡            x                    x                         d        x        +                  1          32                                                                                      ∫                  0                  1                                                                                            ln                      2                                        ⁡                    (                    1                    −                    x                    )                                          ln                      2                                        ⁡                    x                                                        1                    −                    x                                                                   d                x                            ⏟                                                                          1                −                x                                 ↦                                 x                                                                                      =                  1          16                          ∫          0          1                                                    ln              2                        ⁡            (            1            −            x            )                          ln              2                        ⁡            x                    x                         d        x                                          =                  1          8                          ∑                      n            =            1                                ∞                                    (                                    H              n                        n                    −                      1                          n              2                                )                          ∫          0          1                          x                      n            −            1                                    ln          2                ⁡        x                 d        x        =                  1          4                          ∑                      n            =            1                                ∞                                    (                                    H              n                                      n              4                                −                      1                          n              5                                )                                                  =                  1          4                          (          3          ζ          (          5          )          −          ζ          (          2          )          ζ          (          3          )          −          ζ          (          5          )          )                                                  =                  1          4                          (          2          ζ          (          5          )          −          ζ          (          2          )          ζ          (          3          )          )

Show that : int_0^(pi/2) (log^2sin x log^2 cos x)/(cos x sin x)dx=1/4(2zeta(5)-zeta(2)zeta(3))

Answered question

Answer & Explanation

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