Question Number 192715 by mustafazaheen last updated on 25/May/23
$$\int\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{2}} \sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{a}^{\mathrm{2}} }}=? \\ $$
Answered by Frix last updated on 25/May/23
![∫(dx/(x^2 (√(x^2 +a^2 ))))=^([t=x+(√(x^2 +a^2 ))]) =(4/a^2 )∫(t/((t^2 −1)^2 ))dt=−(2/(a^2 (t^2 −1)))= =((x−(√(x^2 +a^2 )))/(a^2 x))+C](https://www.tinkutara.com/question/Q192718.png)
$$\int\frac{{dx}}{{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{2}} +{a}^{\mathrm{2}} }}\overset{\left[{t}={x}+\sqrt{{x}^{\mathrm{2}} +{a}^{\mathrm{2}} }\right]} {=} \\ $$$$=\frac{\mathrm{4}}{{a}^{\mathrm{2}} }\int\frac{{t}}{\left({t}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} }{dt}=−\frac{\mathrm{2}}{{a}^{\mathrm{2}} \left({t}^{\mathrm{2}} −\mathrm{1}\right)}= \\ $$$$=\frac{{x}−\sqrt{{x}^{\mathrm{2}} +{a}^{\mathrm{2}} }}{{a}^{\mathrm{2}} {x}}+{C} \\ $$