Can the following integral be evaluated for x ≠ { 0 , 1 2 }? ∫ 0 1 cos 2 ⁡ ( π x ) x 2 ( 1 − 4 x 2 ) 2 d x

Question

Can the following integral be evaluated for   x  ≠  {  0  ,      1    2    }?      ∫    0          1                          cos                  2                    ⁡      (      π      x      )                      x        2            (      1      −      4              x        2                    )        2                  d    x

Maffei2el · Accepted Answer

If you look at part of the integral, for instance      ∫    0          ε                          cos        2            ⁡      (      π      x      )                      x        2            (      1      −      4              x        2                    )        2              d  x  ,    ε  &amp;lt;      1    2  you&#039;ll see that its convergence is equivalent to the convergence of       ∫    0          ε            1          x      2        d  x, but this last integral is divergent.

Melissa Walker · Accepted Answer

Around x=0,                    cos        2            ⁡      (      π      x      )                      x        2                              (          1          −          4                      x            2                    )                2              =      1          x      2        +      (    8    −          π      2        )    +      (    48    −    8          π      2        +                  π        4            3        )        x    2    +  O      (          x      4        )  and then the problem.By the way, using the 1400 years old approximation  cos  ⁡  (  x  )  ≃                    π        2            −      4              x        2                            π        2            +              x        2                for    −      π    2    ≤  x  ≤      π    2  you have                    cos        2            ⁡      (      π      x      )                      x        2                              (          1          −          4                      x            2                    )                2              ∼      1                  x        2                              (                      x            2                    +          1          )                2            which, probably, makes the problem simpler to analyse.Just for your curiosity, the antiderivative does exist. It is not too difficult since      1                  x        2                              (          1          −          4                      x            2                    )                2              =      1          x      2        +      3          2      x      +      1        −      3          2      x      −      1        +      1          (      2      x      +      1              )        2              +      1          (      2      x      −      1              )        2

Can int_0^1 (cos^2(pi x))/(x^2(1−4x^2)^2)dx be evaluated for x != {0,1/2}?

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