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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
$${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{ln}\left(\mathrm{1}+\frac{{a}^{\mathrm{2}} }{{x}^{\mathrm{2}} }\right){dx} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:\int_{\mathrm{0}} ^{\infty} \:{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){dx} \\ $$$$\left.\mathrm{3}\right){calculate}\:\int_{\mathrm{0}} ^{\infty} \:{ln}\left(\mathrm{1}+\frac{\mathrm{2}}{{x}^{\mathrm{2}} }\right){dx} \\ $$
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).
$$\left.\mathrm{1}\right)\:{we}\:{have}\:{f}^{'} \left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\frac{\mathrm{2}{a}}{{x}^{\mathrm{2}} }}{\mathrm{1}+\frac{{a}^{\mathrm{2}} }{{x}^{\mathrm{2}} }}{dx}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{2}{a}}{{x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} }{dx} \\ $$$$=_{{x}=\mid{a}\mid{t}} \:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{2}{a}}{{a}^{\mathrm{2}} \left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\:\mid{a}\mid\:{dt}\:=\:\mathrm{2}\xi\left({a}\right)\int_{\mathrm{0}} ^{\infty} \:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} }\:=\mathrm{2}\xi\left({a}\right).\frac{\pi}{\mathrm{2}}\:=\pi\xi\left({a}\right)\:{with} \\ $$$$\xi\left({a}\right)=\mathrm{1}\:{if}\:{a}>\mathrm{0}\:{and}\:\xi\left({a}\right)=−\mathrm{1}\:{if}\:{a}<\mathrm{0}\:\Rightarrow{f}\left({a}\right)=\pi{a}\xi\left({a}\right)+{c} \\ $$$${c}={f}\left(\mathrm{0}\right)\:\Rightarrow{f}\left({a}\right)\:=\pi{a}\:\xi\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\int_{\mathrm{0}} ^{\infty} \:{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){dx}\:={f}\left(\mathrm{1}\right)=\pi \\ $$$$\left.\mathrm{3}\right)\:\int_{\mathrm{0}} ^{\infty} \:{ln}\left(\mathrm{1}+\frac{\mathrm{2}}{{x}^{\mathrm{2}} }\right){dx}\:={f}\left(\sqrt{\mathrm{2}}\right)\:=\pi\sqrt{\mathrm{2}}. \\ $$

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