Question Number 66339 by mathmax by abdo last updated on 12/Aug/19
$${calculate}\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\frac{{dx}}{\:\sqrt{\mathrm{4}{x}^{\mathrm{2}} −\mathrm{1}}+\sqrt{\mathrm{4}{x}^{\mathrm{2}} \:+\mathrm{1}}} \\ $$
Commented by Prithwish sen last updated on 13/Aug/19
![(1/2)∫_(1/2) ^1 [(√(4x^2 +1)) − (√(4x^2 −1])) dx = (1/4)[x(√(4x^2 +1)) +(1/2)ln∣2x+(√(4x^2 +1))∣−x(√(4x^2 −1))+(1/2)ln∣2x+(√(4x^2 −1))∣]_(1/2) ^1](https://www.tinkutara.com/question/Q66359.png)
$$\frac{\mathrm{1}}{\mathrm{2}}\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \left[\sqrt{\mathrm{4x}^{\mathrm{2}} +\mathrm{1}}\:−\:\sqrt{\left.\mathrm{4x}^{\mathrm{2}} −\mathrm{1}\right]}\:\mathrm{dx}\right. \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{x}\sqrt{\mathrm{4x}^{\mathrm{2}} +\mathrm{1}}\:+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\mid\mathrm{2x}+\sqrt{\mathrm{4x}^{\mathrm{2}} +\mathrm{1}}\mid−\mathrm{x}\sqrt{\mathrm{4x}^{\mathrm{2}} −\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\mid\mathrm{2x}+\sqrt{\mathrm{4x}^{\mathrm{2}} −\mathrm{1}}\mid\right]_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \\ $$