Question Number 19615 by icyfalcon999 last updated on 13/Aug/17
Answered by Tinkutara last updated on 13/Aug/17
![(1/4)∫_(−∞) ^∞ (dx/(x^2 + ((1/2))^2 )) = (1/2)[tan^(−1) 2x]_(−∞) ^∞ = (1/2)[(π/2) − (−(π/2))] = (π/2)](https://www.tinkutara.com/question/Q19617.png)
$$\frac{\mathrm{1}}{\mathrm{4}}\underset{−\infty} {\overset{\infty} {\int}}\frac{{dx}}{{x}^{\mathrm{2}} \:+\:\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} }\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{tan}^{−\mathrm{1}} \:\mathrm{2}{x}\right]_{−\infty} ^{\infty} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\left[\frac{\pi}{\mathrm{2}}\:−\:\left(−\frac{\pi}{\mathrm{2}}\right)\right]\:=\:\frac{\pi}{\mathrm{2}} \\ $$