Menu Close

Question-211398




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

Leave a Reply

Your email address will not be published. Required fields are marked *