I have a function f ( x ) = x cos 2 ⁡ ( π x )Find the size of the aria that is trapped betweenthe function f(x) and x axis between [0,1.5] ∫ 0 3 2 x cos 2 ⁡ ( π x )

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I have a function   f  (  x  )  =  x      cos    2    ⁡  (  π  x  )Find the size of the aria that is trapped betweenthe function f(x) and x axis between [0,1.5]      ∫    0          3      2        x      cos    2    ⁡  (  π  x  )

apopihvj · Accepted Answer

The answer is       9    16    −      1          4              π                  2                    Let   f  :  x  ↦  x      cos          2        ⁡  (  π  x  ) and       I    =          ∫              0                    3                  /                2              f    (    x    )          d        x  . The function f is continuous on   [  0  ;  3      /    2  ] so I there exist and is finite.Note that  ∀  x  ∈      R    :        cos          2        ⁡  (  π  x  )  =      1    2        (    cos    ⁡    (    2    π    x    )    +    1    )    .Then,                    ∫        x                  cos                      2                          ⁡        (        π        x        )                  d                x                            =                              1          2                ∫        x        (        cos        ⁡        (        2        π        x        )        +        1        )                  d                x                                          =                              1          2                ∫        x        cos        ⁡        2        π        x                  d                x        +                  1          2                ∫                  d                x                                                    =                      I            B            P                                                            x            sin            ⁡            2            π            x                                4            π                          −                  1                      4            π                          ∫        sin        ⁡        2        π        x                  d                x        +                              x                          2                                4                                                            =                      t            ↦            2            π            x                                                            x            sin            ⁡            2            π            x                                4            π                          −                  1                      8                          π                              2                                                    ∫        sin        ⁡        t                  d                t        +                              x                          2                                4                                                  =                                          cos            ⁡            t                                8                          π                              2                                                    +                              x            sin            ⁡            2            π            x                                4            π                          +                              x                          2                                4                +        C                                          =                                          2            π            x            (            π            x            +            sin            ⁡            (            2            π            x            )            )            +            cos            ⁡            (            2            π            x            )                                8                          π                              2                                                    +        C            Therefore,                                                        ∫                              0                                            3                                  /                                2                                      x                          cos                              2                                      ⁡            (            π            x            )                          d                        x            =                          9              16                        −                          1                              4                                  π                                      2                                                                        ≈            0.537                                .

Jaiden Elliott · Accepted Answer

You could try to use   c  o      s    2    (  π  x  )  =            c      o      s      (      2      π      x      )      +      1        2   before applying integration by parts.

what is the integral of int x cos^2(pi x)

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