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lim-x-0-1-1-t-2-3-cos-xt-dt-

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Question Number 133091 by metamorfose last updated on 18/Feb/21
lim_(x→+∞) ∫_0 ^1 (1−t^2 )^3 cos(xt)dt...?
limx→+∞∫01(1−t2)3cos(xt)dt…?
Answered by mnjuly1970 last updated on 18/Feb/21
answer:=0 reiman −lebesgue theorem f is continuoues on[0,1] lim_(x→∞) ∫_0 ^( 1) f(t)sin(xt)dt=0
answer:=0reiman−lebesguetheoremfiscontinuoueson[0,1]limx→∞∫01f(t)sin(xt)dt=0
Answered by mathmax by abdo last updated on 20/Feb/21
let f(t) is pritive of (1−t^2 )^3 by parts we get ∫_0 ^1 (1−t^2 )^3 cos(xt)dt =[((f(t))/x) sin(xt)]_0 ^1 −∫_0 ^1 ((f(t))/x)sin(xt)dt =((f(1)sin(x))/x) −(1/x)∫_0 ^1 f(t)sin(xt)dt ⇒ for x>0 ∣∫_0 ^1 (...)dt∣ ≤((∣f(1)∣)/x) +(1/x)∫_0 ^1 ∣f(t)∣ dt →0 (x→+∞) ⇒ lim_(x→+∞) ∫_0 ^1 (1−t^2 )^3 cos(xt)dt =0
letf(t)ispritiveof(1−t2)3bypartsweget∫01(1−t2)3cos(xt)dt=[f(t)xsin(xt)]01−∫01f(t)xsin(xt)dt=f(1)sin(x)x−1x∫01f(t)sin(xt)dt⇒forx>0∣∫01(…)dt∣⩽∣f(1)∣x+1x∫01∣f(t)∣dt→0(x→+∞)⇒limx→+∞∫01(1−t2)3cos(xt)dt=0

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