The integral of | cos ⁡ x x | I'm looking to determine whether the following function is unbounded or not: F ( x ) = ∫ 1 x | cos ⁡ t t | d tI can't seem to do much with it because of the | cos ⁡ ( t ) | . I thought of using the fact that ∫ | f | ≥ | ∫ f | , but the problem is that the integral of cos ⁡ t t (without the absolute values) is bounded, and so that doesn't prove that F(x) is unbounded or bounded. I tried re-expressing this as a cosine integral (the function Ci(x)) but to no avail. I'm not sure where else to go with this; the main problem seems to be the fact that its very difficult to derive an inequality with the | cos ⁡ ( t ) | without a | cos ⁡ ( t ) | on the other side of the inequality (or at least some trig function).

Question

The integral of       |                  cos        ⁡        x            x        |  I&#039;m looking to determine whether the following function is unbounded or not:  F  (  x  )  =      ∫    1    x        |                  cos        ⁡        t            t        |    d  tI can&#039;t seem to do much with it because of the       |    cos  ⁡  (  t  )      |  . I thought of using the fact that   ∫      |    f      |    ≥      |    ∫  f      |  , but the problem is that the integral of             cos      ⁡      t        t   (without the absolute values) is bounded, and so that doesn&#039;t prove that F(x) is unbounded or bounded. I tried re-expressing this as a cosine integral (the function Ci(x)) but to no avail. I&#039;m not sure where else to go with this; the main problem seems to be the fact that its very difficult to derive an inequality with the       |    cos  ⁡  (  t  )      |   without a       |    cos  ⁡  (  t  )      |   on the other side of the inequality (or at least some trig function).

Turynka2f · Accepted Answer

Hint:Consider the harmonic series and      ∫          π              /            2      +      k      π              3      π              /            2      +      k      π                          |            cos      ⁡      t              |              t      d  t  ⩾      1          3      π              /            2      +      k      π            ∫          π              /            2      +      k      π              3      π              /            2      +      k      π            |    cos  ⁡  t      |      d  t  =      2          3      π              /            2      +      k      π

The integral of | cos ⁡ x </mrow>

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