Finding f ( x ) in cos 2 ⁡ ( x ) f ( x ) = x 2 − 2 ∫ 1 x sin ⁡ ( t ) cos ⁡ ( t ) f ( t ) d t

Question

Finding   f  (  x  ) in       cos    2    ⁡  (  x  )  f  (  x  )  =      x    2    −  2      ∫    1    x    sin  ⁡  (  t  )  cos  ⁡  (  t  )  f  (  t  )        d    t

Oliver Shepherd · Accepted Answer

Assuming your equation reads   f  (  x  )      cos    2    ⁡  x  =      x    2    −  2      ∫    1          x        sin  ⁡  t  cos  ⁡  t  f  (  t  )        d    t then differentiating gives  −  2  f  (  x  )  sin  ⁡  x  cos  ⁡  x  +      cos    2    ⁡  x      f    ′    (  x  )  =  2  x  −  2  f  (  x  )  sin  ⁡  x  cos  ⁡  xso upon simplification, this gives us       f    ′    (  x  )      cos    2    ⁡  x  =  2  x which is a differential equation you can solve  f  (  x  )  =  ∫            2      x                      cos        2            ⁡      x              d    x  =  ∫  2  x      sec    2    ⁡  x        d    xusing IBP or some other technique you fancy. In particular, IBP gives  ∫  2  x      sec    2    ⁡  x        d    x  =  2  x  tan  ⁡  x  −  ∫  2  tan  ⁡  x        d    x

Finding f ( x ) in cos 2 </msup> ⁡ ( x

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