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Question Number 74800 by mathmax by abdo last updated on 30/Nov/19

study the existence of f(x)=∫_0 ^∞ ((tcos(tx))/(1+t^2 ))dt

studytheexistenceoff(x)=∫0∞tcos(tx)1+t2dt

Answered by mind is power last updated on 30/Nov/19

x=0 ∫_0 ^(+∞) (t/(1+t^2 ))dt divege for x#0 ipp [_0 ^(+∞) (t/(t^2 +1)).((sin(xt))/x)]−(1/x)∫_0 ^(+∞) sin(xt).(((1−t^2 ))/((1+t^2 )^2 ))dt =−(1/x)∫_0 ^(+∞) (((1−t^2 )sin(xt))/((1+t^2 )^2 ))dt exist cv abdolutly ≤(1/(∣x∣))∫_0 ^(+∞) ((∣(1−t^2 )∣)/((1+t^2 )^2 ))dt exist since at[+∞ (((t^2 −1))/((t^2 +1)^2 ))∼(1/t^2 ),wich is Reimann/integrabl at +∞ f(x) exist overR^∗

x=0∫0+∞t1+t2dtdivegeYou can't use 'macro parameter character #' in math modeipp[0+∞tt2+1.sin(xt)x]−1x∫0+∞sin(xt).(1−t2)(1+t2)2dt=−1x∫0+∞(1−t2)sin(xt)(1+t2)2dtexistcvabdolutly⩽1∣x∣∫0+∞∣(1−t2)∣(1+t2)2dtexistsinceat[+∞(t2−1)(t2+1)2∼1t2,wichisReimann/integrablat+∞f(x)existoverR∗

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