Question Number 129212 by bramlexs22 last updated on 13/Jan/21
$$\:\mathrm{M}\:=\:\int\:\sqrt{\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{5}}}\:\mathrm{dx}\:? \\ $$
Answered by TheSupreme last updated on 13/Jan/21
$${x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{5}}={t} \\ $$
Answered by MJS_new last updated on 13/Jan/21
![∫(√(x+(√(x^2 +5)))) dx= [t=(√(x+(√(x^2 +5)))) → dx=((t^4 +5)/t^3 )dt] =∫((t^4 +5)/t^2 )dt=((t^4 −15)/(3t))= =((2(2x−(√(x^2 +5)))(√(x+(√(x^2 +5)))))/3)+C](Q129216.png)
$$\int\sqrt{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{5}}}\:{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{5}}}\:\rightarrow\:{dx}=\frac{{t}^{\mathrm{4}} +\mathrm{5}}{{t}^{\mathrm{3}} }{dt}\right] \\ $$$$=\int\frac{{t}^{\mathrm{4}} +\mathrm{5}}{{t}^{\mathrm{2}} }{dt}=\frac{{t}^{\mathrm{4}} −\mathrm{15}}{\mathrm{3}{t}}= \\ $$$$=\frac{\mathrm{2}\left(\mathrm{2}{x}−\sqrt{{x}^{\mathrm{2}} +\mathrm{5}}\right)\sqrt{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{5}}}}{\mathrm{3}}+{C} \\ $$