Question Number 211398 by Spillover last updated on 08/Sep/24
Answered by Frix last updated on 08/Sep/24
$$\int\frac{{dx}}{\left({x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{2}}}\:\overset{{t}=\frac{{x}+\mathrm{1}+\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{2}}}{\:\sqrt{\mathrm{2}}}} {=} \\ $$$$=\mathrm{2}\sqrt{\mathrm{2}}\int\frac{{dt}}{\mathrm{2}{t}^{\mathrm{2}} +\mathrm{1}}=\mathrm{2tan}^{−\mathrm{1}} \:\sqrt{\mathrm{2}}{t}\:= \\ $$$$=\mathrm{2tan}^{−\mathrm{1}} \:\left({x}+\mathrm{1}+\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{2}}\right)\:+{C} \\ $$