Definite integration evaluation of ∫ 0 π x ( a 2 cos 2 ⁡ x + b 2 sin 2 ⁡ x ) 2 d x

Question

Definite integration evaluation of       ∫          0              π            x          (              a        2                    cos        2            ⁡      x      +              b        2                    sin        2            ⁡      x              )        2              d  x

Salma Baird · Accepted Answer

I prefer to the following method:                    I        :=                  ∫                      0                                π                                    x                      (                          a              2                                      cos              2                        ⁡            x            +                          b              2                                      sin              2                        ⁡            x                          )              2                                              d        x                            =                  π          2                          ∫                      0                                π                                                d            x                                (                          a              2                                      cos              2                        ⁡            x            +                          b              2                                      sin              2                        ⁡            x                          )              2                                                                        =        π                  ∫                      0                                              π              2                                                            d            x                                (                          a              2                                      cos              2                        ⁡            x            +                          b              2                                      sin              2                        ⁡            x                          )              2                                                                        =        π                  ∫                      0                                              π              2                                                            1            +                          tan              2                        ⁡            x                                (                          a              2                        +                          b              2                                      tan              2                        ⁡            x                          )              2                                                        sec          2                ⁡        x                d        x        .            Now we make the substitution   b  tan  ⁡  x  ↦  a  tan  ⁡  x. Then                    I                            =                  π                      (            a            b                          )              3                                                ∫                      0                                              π              2                                                                          b              2                        +                          a              2                                      tan              2                        ⁡            x                                (            1            +                          tan              2                        ⁡            x                          )              2                                                        sec          2                ⁡        x                d        x                                          =                  π                      (            a            b                          )              3                                                ∫                      0                                              π              2                                      (                  b          2                          cos          2                ⁡        x        +                  a          2                          sin          2                ⁡        x        )                d        x                                          =                  π                      (            a            b                          )              3                                      ⋅                  π          4                          (                      a            2                    +                      b            2                    )                =                                            π              2                        (                          a              2                        +                          b              2                        )                                4            (            a            b                          )              3                                      .

Definite integration evaluation of int_0^pi x/(a^2 cos^2x+b^2 sin^2 x)^2dx

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