Question Number 175467 by manish54 last updated on 31/Aug/22
Commented by Frix last updated on 01/Sep/22
![∫_1 ^∞ (dx/((x^2 +1)^(1/2) ))=[sinh x]_1 ^∞ does not converge ∫_1 ^∞ (dx/((x^2 +1)^(3/2) ))=[(x/((x^2 +1)^(1/2) ))]_1 ^∞ =1−(1/( (√2)))](Q175515.png)
$$\underset{\mathrm{1}} {\overset{\infty} {\int}}\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} }=\left[\mathrm{sinh}\:{x}\right]_{\mathrm{1}} ^{\infty} \:\mathrm{does}\:\mathrm{not}\:\mathrm{converge} \\ $$$$\underset{\mathrm{1}} {\overset{\infty} {\int}}\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}/\mathrm{2}} }=\left[\frac{{x}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} }\right]_{\mathrm{1}} ^{\infty} =\mathrm{1}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}} \\ $$