Question Number 103741 by mathmax by abdo last updated on 17/Jul/20

calculate ∫_0 ^∞   ((arctan(ch(x)))/(x^2  +9))dx

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

I =∫_0 ^∞  ((arctan(chx))/(x^2 +9)) dx ⇒2A = ∫_(−∞) ^(+∞)  ((arctan(chx))/(x^2  +9))dx let ϕ(z) =((arctan(chz))/(z^2 +9))  we have lim_(z→+∞) ∣zϕ(z)∣=0 and ϕ(z) =((arctan(chz))/((z−3i)(z+3i)))  residus theorem give ∫_(−∞) ^(+∞)  ϕ(z)dz =2iπ Res(ϕ,3i)  =2iπ × ((arctan(ch(3i)))/(6i)) =(π/3) arctan(((e^(3i) +e^(−3i) )/2)) =(π/3) arctan(cos(3)) ⇒  A =(π/6) arctan(cos3)