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Question Number 108583 by mathmax by abdo last updated on 17/Aug/20

find ∫_0 ^1  ((ln(1+x^2 ))/(1+x^2 ))dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$

Answered by mnjuly1970 last updated on 18/Aug/20

subs:x=tan(t)  ⇒Ω=−2∫_0 ^( (π/4)) ln(cos(t))dt=−2((G/2)−(π/4)ln(2))  Ω=(π/2)ln(2)−G ....G is  catalan  constant....

$$\mathrm{subs}:\mathrm{x}=\mathrm{tan}\left(\mathrm{t}\right) \\ $$$$\Rightarrow\Omega=−\mathrm{2}\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \mathrm{ln}\left(\mathrm{cos}\left(\mathrm{t}\right)\right)\mathrm{dt}=−\mathrm{2}\left(\frac{\mathrm{G}}{\mathrm{2}}−\frac{\pi}{\mathrm{4}}\mathrm{ln}\left(\mathrm{2}\right)\right) \\ $$$$\Omega=\frac{\pi}{\mathrm{2}}\mathrm{ln}\left(\mathrm{2}\right)−\mathrm{G}\:....\mathrm{G}\:\mathrm{is}\:\:\mathrm{catalan} \\ $$$$\mathrm{constant}.... \\ $$

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