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Question Number 44305 by abdo.msup.com last updated on 26/Sep/18
let f(a) =∫_0 ^∞ ln(1+(a^2 /x^2 ))dx 1) find a explicit form of f(x) 2)find ∫_0 ^∞ ln(1+(1/x^2 ))dx 3)calculate ∫_0 ^∞ ln(1+(2/x^2 ))dx
letf(a)=∫0∞ln(1+a2x2)dx1)findaexplicitformoff(x)2)find∫0∞ln(1+1x2)dx3)calculate∫0∞ln(1+2x2)dx
Commented by maxmathsup by imad last updated on 29/Sep/18
1) we have f^′ (a) =∫_0 ^∞ (((2a)/x^2 )/(1+(a^2 /x^2 )))dx = ∫_0 ^∞ ((2a)/(x^2 +a^2 ))dx =_(x=∣a∣t) ∫_0 ^∞ ((2a)/(a^2 (1+t^2 ))) ∣a∣ dt = 2ξ(a)∫_0 ^∞ (dt/(1+t^2 )) =2ξ(a).(π/2) =πξ(a) with ξ(a)=1 if a>0 and ξ(a)=−1 if a<0 ⇒f(a)=πaξ(a)+c c=f(0) ⇒f(a) =πa ξ(a) 2)∫_0 ^∞ ln(1+(1/x^2 ))dx =f(1)=π 3) ∫_0 ^∞ ln(1+(2/x^2 ))dx =f((√2)) =π(√2).
1)wehavef′(a)=∫0∞2ax21+a2x2dx=∫0∞2ax2+a2dx=x=∣a∣t∫0∞2aa2(1+t2)∣a∣dt=2ξ(a)∫0∞dt1+t2=2ξ(a).π2=πξ(a)withξ(a)=1ifa>0andξ(a)=−1ifa<0⇒f(a)=πaξ(a)+cc=f(0)⇒f(a)=πaξ(a)2)∫0∞ln(1+1x2)dx=f(1)=π3)∫0∞ln(1+2x2)dx=f(2)=π2.

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