Finding f(x) in cos 2 </msup> &#x2061;<!-- ⁡ --> ( x ) f ( x ) =

velitshh

velitshh

Answered question

2022-05-27

Finding f(x) in cos 2 ( x ) f ( x ) = x 2 2 1 x sin ( t ) cos ( t ) f ( t ) d t
I can apply the FToC and I get:
( 2 cos ( x ) sin ( x ) f ( x ) ) + ( cos 2 x f ( x ) ) = 2 x sin ( x ) cos ( x ) f ( x )
Now I don't understand how to find f(x).

Answer & Explanation

asafand2c

asafand2c

Beginner2022-05-28Added 11 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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