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kolutastmr 2022-07-02 Answered
Finding f ( x ) in cos 2 ( x ) f ( x ) = x 2 2 1 x sin ( t ) cos ( t ) f ( t ) d t
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Answers (1)

Oliver Shepherd
Answered 2022-07-03 Author has 24 answers
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 x
so 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 x
using 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
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