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0-3-xdx-x-3-2x-2-x-2-

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Question Number 163472 by Zaynal last updated on 07/Jan/22
∫_0 ^3 ((xdx)/(x^3 + 2x^2 + x 2)) =
$$\int_{\mathrm{0}} ^{\mathrm{3}} \:\frac{\boldsymbol{\mathrm{xdx}}}{\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:+\:\mathrm{2}\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:+\:\boldsymbol{\mathrm{x}}\:\mathrm{2}}\:= \\ $$
Answered by alephzero last updated on 07/Jan/22
∫_0 ^3 (x/(x^3 +2x^2 +2x))dx = lim_(a→0^+ ) (∫_a ^3 (x/(x^3 +2x^2 +2x))dx) = ∫(x/(x^3 +2x^2 +2x))dx = ∫(x/(x(x^2 +2x+2)))dx = = ∫(dx/(x^2 +2x+2)) = ∫(dx/(x^2 +2x+1+1)) = = ∫(dx/((x+1)^2 +1)) Let t = x+1 & dx = dt ⇒ ∫(dt/(t^2 +1)) = (1/1)arctan(t/1) = arctan t = arctan(x+1) ∫_a ^3 (x/(x^3 +2x^2 +2x))dx = arctan(3+1)− −arctan(a+1) ⇒lim_(a→0^+ ) (∫_a ^3 (x/(x^3 +2x^2 +2x))dx) = = arctan(4) − arctan(0+1) = = arctan(4) − arctan(1) = = arctan(4)−(π/4) (Or ≈ 0.54)
$$\int_{\mathrm{0}} ^{\mathrm{3}} \frac{{x}}{{x}^{\mathrm{3}} +\mathrm{2}{x}^{\mathrm{2}} +\mathrm{2}{x}}{dx}\:=\:\underset{{a}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\left(\int_{{a}} ^{\mathrm{3}} \frac{{x}}{{x}^{\mathrm{3}} +\mathrm{2}{x}^{\mathrm{2}} +\mathrm{2}{x}}{dx}\right)\:= \\ $$$$\int\frac{{x}}{{x}^{\mathrm{3}} +\mathrm{2}{x}^{\mathrm{2}} +\mathrm{2}{x}}{dx}\:=\:\int\frac{\cancel{{x}}}{\cancel{{x}}\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}\right)}{dx}\:=\: \\ $$$$=\:\int\frac{{dx}}{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}}\:=\:\int\frac{{dx}}{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}+\mathrm{1}}\:= \\ $$$$=\:\int\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{1}} \\ $$$$\mathrm{Let}\:{t}\:=\:{x}+\mathrm{1}\:\&\:{dx}\:=\:{dt} \\ $$$$\Rightarrow\:\int\frac{{dt}}{{t}^{\mathrm{2}} +\mathrm{1}}\:=\:\frac{\mathrm{1}}{\mathrm{1}}\mathrm{arctan}\frac{{t}}{\mathrm{1}}\:=\:\mathrm{arctan}\:{t}\:= \\ $$$$\mathrm{arctan}\left({x}+\mathrm{1}\right) \\ $$$$\int_{{a}} ^{\mathrm{3}} \frac{{x}}{{x}^{\mathrm{3}} +\mathrm{2}{x}^{\mathrm{2}} +\mathrm{2}{x}}{dx}\:=\:\mathrm{arctan}\left(\mathrm{3}+\mathrm{1}\right)− \\ $$$$−\mathrm{arctan}\left({a}+\mathrm{1}\right) \\ $$$$\Rightarrow\underset{{a}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\left(\int_{{a}} ^{\mathrm{3}} \frac{{x}}{{x}^{\mathrm{3}} +\mathrm{2}{x}^{\mathrm{2}} +\mathrm{2}{x}}{dx}\right)\:=\: \\ $$$$=\:\mathrm{arctan}\left(\mathrm{4}\right)\:−\:\mathrm{arctan}\left(\mathrm{0}+\mathrm{1}\right)\:=\: \\ $$$$=\:\mathrm{arctan}\left(\mathrm{4}\right)\:−\:\mathrm{arctan}\left(\mathrm{1}\right)\:=\: \\ $$$$=\:\mathrm{arctan}\left(\mathrm{4}\right)−\frac{\pi}{\mathrm{4}}\: \\ $$$$\left(\mathrm{Or}\:\approx\:\mathrm{0}.\mathrm{54}\right) \\ $$

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