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Question Number 159474 by mnjuly1970 last updated on 17/Nov/21

        nice  integral.    prove  that  ∫^( ∞) _0 (( tan^( −1) (2x)+ tan^( −1) ((x/2) ))/(1+x^( 2) ))dx=(π^( 2) /4)

$$ \\ $$$$\:\:\:\:\:\:{nice}\:\:{integral}. \\ $$$$\:\:{prove}\:\:{that} \\ $$$$\underset{\mathrm{0}} {\int}^{\:\infty} \frac{\:{tan}^{\:−\mathrm{1}} \left(\mathrm{2}{x}\right)+\:{tan}^{\:−\mathrm{1}} \left(\frac{{x}}{\mathrm{2}}\:\right)}{\mathrm{1}+{x}^{\:\mathrm{2}} }{dx}=\frac{\pi^{\:\mathrm{2}} }{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$ \\ $$

Answered by qaz last updated on 18/Nov/21

∫_0 ^∞ ((tan^(−1) (2x)+tan^(−1) ((x/2)))/(1+x^2 ))dx  =∫_0 ^∞ ((tan^(−1) (2/x)+tan^(−1) (x/2))/(1+x^2 ))dx.....2x→(2/x)  =(π/2)∫_0 ^∞ (dx/(1+x^2 ))  =(π^2 /4)

$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{2x}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$$$=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{2}}{\mathrm{x}}+\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{x}}{\mathrm{2}}}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx}.....\mathrm{2x}\rightarrow\frac{\mathrm{2}}{\mathrm{x}} \\ $$$$=\frac{\pi}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{dx}}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} } \\ $$$$=\frac{\pi^{\mathrm{2}} }{\mathrm{4}} \\ $$

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