Show that int_0^(oo) 1/t (cos(at)−cos(bt))dt=ln(b/a),a,b>0.

oliadas73 2022-09-05 Answered
Show that 0 1 t ( cos ( a t ) cos ( b t ) ) d t = ln ( b / a ) , a , b > 0
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Answers (1)

Yuliana Griffith
Answered 2022-09-06 Author has 6 answers
Write
1 t = 0 d p e p t
Then the integral is
0 d t cos a t cos b t t = 0 d t ( cos a t cos b t ) 0 d p e p t = 0 d p ( 0 d t cos a t e p t 0 d t cos b t e p t ) = 0 d p ( p p 2 + a 2 p p 2 + b 2 )
It seems you got the rest. Note that, in the second line, we can reverse the order of integration because the integrals involved are convergent.
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