Prove int_0^infty sin^4x/x^4 dx=pi/3

Demarion Ortega 2022-11-20 Answered
Prove 0 sin 4 x x 4 d x = π 3
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Answers (2)

Zoe Andersen
Answered 2022-11-21 Author has 16 answers
You are likely expected to integrate by parts (twice)
sin 4 ( x ) x 4 d x = 1 3 sin 4 ( x ) x 3 + 4 3 cos ( x ) sin 3 ( x ) x 3 d x = 1 3 sin 4 ( x ) x 3 2 cos ( x ) sin 3 ( x ) 3 x 2 + 2 3 3 cos 2 ( x ) sin 2 ( x ) sin 4 ( x ) x 2 d x = 1 3 sin 4 ( x ) x 3 2 cos ( x ) sin 3 ( x ) 3 x 2 + 2 3 ( sin 2 ( 2 x ) x 2 sin 2 ( x ) x 2 ) d x
where the last equality used
3 cos 2 ( x ) sin 2 ( x ) sin 4 ( x ) = 3 cos 2 ( x ) sin 2 ( x ) sin 2 ( x ) ( 1 cos 2 ( x ) ) = ( 2 sin ( x ) cos ( x ) ) 2 sin 2 ( x ) = sin 2 ( 2 x ) sin 2 ( x )
Now
0 sin 4 ( x ) x 4 d x = 2 3 0 sin 2 ( 2 x ) x 2 d x 2 3 0 sin 2 ( x ) x 2 d x = 4 3 0 sin 2 ( y ) y 2 d y 2 3 0 sin 2 ( x ) x 2 d x = 2 3 0 sin 2 ( x ) x 2 d x = π 3
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evitagimm9h
Answered 2022-11-22 Author has 5 answers
That is, the FT of ( sin x / x ) 2 is
d x sin 2 x x 2 e i k x = { π ( 1 | k | 2 ) | k | 2 0 | k | > 2
Plancherel/Parseval says that
d x sin 4 x x 4 = 1 2 π 2 2 d k π 2 ( 1 | k | 2 ) 2 = π 2 4 3 = 2 π 3
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