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Question Number 108502 by mnjuly1970 last updated on 17/Aug/20

Commented by mnjuly1970 last updated on 17/Aug/20

♣please prove  above integrals  equalities.♣

pleaseproveaboveintegralsequalities.

Answered by mathmax by abdo last updated on 17/Aug/20

1)  A_p =∫_0 ^(π/2)  sin(ptanx)tanx dx  changement  tanx =t give  A_p =∫_0 ^∞ sin(pt)t (dt/(1+t^2 )) =∫_0 ^∞   ((tsin(pt))/(1+t^2 )) dt  =(1/2)∫_(−∞) ^(+∞)  ((tsin(pt))/(t^2  +1))dt  =(1/2)Im(∫_(−∞) ^(+∞ )  ((te^(ipt) )/(t^2  +1))dt)  let  ϕ(z) =((z e^(ipz) )/(z^2  +1)) ⇒ϕ(z) =((z e^(ipz) )/((z−i)(z+i)))  residus theorem give  ∫_(−∞) ^(+∞)  ϕ(z)dz =2iπRes(ϕ,i) =2iπ  .((i e^(−p) )/(2i)) =iπ e^(−p)  ⇒  ★A_p =(π/2)e^(−p)  ★

1)Ap=0π2sin(ptanx)tanxdxchangementtanx=tgiveAp=0sin(pt)tdt1+t2=0tsin(pt)1+t2dt=12+tsin(pt)t2+1dt=12Im(+teiptt2+1dt)letφ(z)=zeipzz2+1φ(z)=zeipz(zi)(z+i)residustheoremgive+φ(z)dz=2iπRes(φ,i)=2iπ.iep2i=iπepAp=π2ep

Commented by mnjuly1970 last updated on 17/Aug/20

thank you sir

thankyousir

Commented by mathmax by abdo last updated on 17/Aug/20

you are welcome

youarewelcome

Commented by mnjuly1970 last updated on 17/Aug/20

the god bless you and keep you my  friend and my masster ..(thank you)^∞

thegodblessyouandkeepyoumyfriendandmymasster..(thankyou)

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