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Question Number 46156 by Meritguide1234 last updated on 21/Oct/18

	Answered by tanmay.chaudhury50@gmail.com last updated on 21/Oct/18
	$x^4 +2x^3 −x^2 +2x+1 =x^2 (x^2 +(1/x^2 )+2x+(2/x)−1) =x^2 {(x+(1/x))^2 −2+2(x+(1/x))−1} =x^2 {(x+(1/x))^2 +2(x+(1/x))−3} ∫(((1−(1/x^2 ))×x(√({(x+(1/x))^2 +2(x+(1/x))−3)) )/((x+1)^2 )) dx ∫(((1−(1/x^2 ))×x×(√({(x+(1/x))^2 +2(x+(1/x))−3)) dx)/(x(x+2+(1/x)))) t=x+(1/x) dt=1−(1/x^2 )dx ∫((√(t^2 +2t−3))/(t+2))dt ←standard form ∫((t^2 +2t−3)/((t+2)(√(t^2 +2t−3)) ))dt ∫((tdt)/(√(t^2 +2t−3)))−∫((3dt)/((t+2)(√(t^2 +2t−3)))) (1/2)∫((2t+2−2)/(√(t^2 +2t−3)))−3∫(dt/((t+2)(√(t^2 +2t−3)) )) (1/2)∫((d(t^2 +2t−3))/(√(t^2 +2t−3)))−∫(dt/(√((t+1)^2 −4)))−∫(dt/((t+2)(√(t^2 +2t−3)))) (1/2)×(((t^2 +2t−3)^(((−1)/2)+1) )/(1/2))−ln{(t+1)+(√(t^2 +2t−1)) +I_3 ={(x+(1/x))^2 +2(x+(1/x))−3}^(1/2) −ln{(x+(1/x)+1)+(√((x+(1/x))^2 +2(x+(1/x))1)) +I_3 now calcukating I_3 ∫(dt/((t+2)(√(t^2 +2t−3)))) t+2=(1/k) dt=((−1)/k^2 )dk ∫((−dk)/(k^2 ×(1/k)(√(((1/k)−2)^2 +2((1/k)−2)−3)))) ∫((−dk)/(k(√((1/k^2 )−(4/k)+4+(2/k)−4−3)))) ∫((−dk)/(k(√((1−4k+4k^2 +2k−7k^2 )/k^2 )))) ∫((−dk)/(√(−3k^2 −2k+1))) ∫((−dk)/(√(1−3(k^2 +(2/3)k+(1/9)−(1/9))))) ∫((−dk)/(√(1−3{(k+(1/3))^2 −(1/9)}))) ∫((−dk)/(√(1+(1/3)−3(k+(1/3))^2 ))) ∫((−dk)/(√((4/3)−3(k+(1/3))^2 ))) (1/(√3))∫((−dk)/(√(((2/3))^2 −(k+(1/3))^2 ))) =((−1)/(√3))×sin^(−1) (((k+(1/3))/(2/3))) =((−1)/(√3))sin^(−1) ((((1/(t+2))+(1/3))/(2/3))) ((−1)/(√3))sin^(−1) ((((1/(x+(1/x)+2))+(1/3))/(2/3)))+c$
	$${x}^{\mathrm{4}} +\mathrm{2}{x}^{\mathrm{3}} −{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1} \\ $$$$={x}^{\mathrm{2}} \left({x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}^{\mathrm{2}} }+\mathrm{2}{x}+\frac{\mathrm{2}}{{x}}−\mathrm{1}\right) \\ $$$$={x}^{\mathrm{2}} \left\{\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} −\mathrm{2}+\mathrm{2}\left({x}+\frac{\mathrm{1}}{{x}}\right)−\mathrm{1}\right\} \\ $$$$={x}^{\mathrm{2}} \left\{\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} +\mathrm{2}\left({x}+\frac{\mathrm{1}}{{x}}\right)−\mathrm{3}\right\} \\ $$$$\int\frac{\left(\mathrm{1}−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)×{x}\sqrt{\left\{\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} +\mathrm{2}\left({x}+\frac{\mathrm{1}}{{x}}\right)−\mathrm{3}\right.}\:}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} }\:{dx} \\ $$$$\int\frac{\left(\mathrm{1}−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)×{x}×\sqrt{\left\{\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} +\mathrm{2}\left({x}+\frac{\mathrm{1}}{{x}}\right)−\mathrm{3}\right.}\:{dx}}{{x}\left({x}+\mathrm{2}+\frac{\mathrm{1}}{{x}}\right)} \\ $$$${t}={x}+\frac{\mathrm{1}}{{x}}\:\:\:{dt}=\mathrm{1}−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }{dx} \\ $$$$\int\frac{\sqrt{{t}^{\mathrm{2}} +\mathrm{2}{t}−\mathrm{3}}}{{t}+\mathrm{2}}{dt}\:\:\:\leftarrow{standard}\:{form}\: \\ $$$$ \\ $$$$\int\frac{{t}^{\mathrm{2}} +\mathrm{2}{t}−\mathrm{3}}{\left({t}+\mathrm{2}\right)\sqrt{{t}^{\mathrm{2}} +\mathrm{2}{t}−\mathrm{3}}\:}{dt} \\ $$$$ \\ $$$$\int\frac{{tdt}}{\sqrt{{t}^{\mathrm{2}} +\mathrm{2}{t}−\mathrm{3}}}−\int\frac{\mathrm{3}{dt}}{\left({t}+\mathrm{2}\right)\sqrt{{t}^{\mathrm{2}} +\mathrm{2}{t}−\mathrm{3}}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{2}{t}+\mathrm{2}−\mathrm{2}}{\sqrt{{t}^{\mathrm{2}} +\mathrm{2}{t}−\mathrm{3}}}−\mathrm{3}\int\frac{{dt}}{\left({t}+\mathrm{2}\right)\sqrt{{t}^{\mathrm{2}} +\mathrm{2}{t}−\mathrm{3}}\:} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{d}\left({t}^{\mathrm{2}} +\mathrm{2}{t}−\mathrm{3}\right)}{\sqrt{{t}^{\mathrm{2}} +\mathrm{2}{t}−\mathrm{3}}}−\int\frac{{dt}}{\sqrt{\left({t}+\mathrm{1}\right)^{\mathrm{2}} −\mathrm{4}}}−\int\frac{{dt}}{\left({t}+\mathrm{2}\right)\sqrt{{t}^{\mathrm{2}} +\mathrm{2}{t}−\mathrm{3}}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}×\frac{\left({t}^{\mathrm{2}} +\mathrm{2}{t}−\mathrm{3}\right)^{\frac{−\mathrm{1}}{\mathrm{2}}+\mathrm{1}} }{\frac{\mathrm{1}}{\mathrm{2}}}−{ln}\left\{\left({t}+\mathrm{1}\right)+\sqrt{{t}^{\mathrm{2}} +\mathrm{2}{t}−\mathrm{1}}\:+{I}_{\mathrm{3}} \right. \\ $$$$=\left\{\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} +\mathrm{2}\left({x}+\frac{\mathrm{1}}{{x}}\right)−\mathrm{3}\right\}^{\frac{\mathrm{1}}{\mathrm{2}}} −{ln}\left\{\left({x}+\frac{\mathrm{1}}{{x}}+\mathrm{1}\right)+\sqrt{\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} +\mathrm{2}\left({x}+\frac{\mathrm{1}}{{x}}\right)\mathrm{1}}\:+{I}_{\mathrm{3}} \right. \\ $$$${now}\:{calcukating}\:{I}_{\mathrm{3}} \\ $$$$\int\frac{{dt}}{\left({t}+\mathrm{2}\right)\sqrt{{t}^{\mathrm{2}} +\mathrm{2}{t}−\mathrm{3}}} \\ $$$${t}+\mathrm{2}=\frac{\mathrm{1}}{{k}}\:\:\:{dt}=\frac{−\mathrm{1}}{{k}^{\mathrm{2}} }{dk} \\ $$$$\int\frac{−{dk}}{{k}^{\mathrm{2}} ×\frac{\mathrm{1}}{{k}}\sqrt{\left(\frac{\mathrm{1}}{{k}}−\mathrm{2}\right)^{\mathrm{2}} +\mathrm{2}\left(\frac{\mathrm{1}}{{k}}−\mathrm{2}\right)−\mathrm{3}}} \\ $$$$\int\frac{−{dk}}{{k}\sqrt{\frac{\mathrm{1}}{{k}^{\mathrm{2}} }−\frac{\mathrm{4}}{{k}}+\mathrm{4}+\frac{\mathrm{2}}{{k}}−\mathrm{4}−\mathrm{3}}} \\ $$$$\int\frac{−{dk}}{{k}\sqrt{\frac{\mathrm{1}−\mathrm{4}{k}+\mathrm{4}{k}^{\mathrm{2}} +\mathrm{2}{k}−\mathrm{7}{k}^{\mathrm{2}} }{{k}^{\mathrm{2}} }}} \\ $$$$\int\frac{−{dk}}{\sqrt{−\mathrm{3}{k}^{\mathrm{2}} −\mathrm{2}{k}+\mathrm{1}}} \\ $$$$\int\frac{−{dk}}{\sqrt{\mathrm{1}−\mathrm{3}\left({k}^{\mathrm{2}} +\frac{\mathrm{2}}{\mathrm{3}}{k}+\frac{\mathrm{1}}{\mathrm{9}}−\frac{\mathrm{1}}{\mathrm{9}}\right)}} \\ $$$$\int\frac{−{dk}}{\sqrt{\mathrm{1}−\mathrm{3}\left\{\left({k}+\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{9}}\right\}}} \\ $$$$\int\frac{−{dk}}{\sqrt{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}−\mathrm{3}\left({k}+\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{2}} }} \\ $$$$\int\frac{−{dk}}{\sqrt{\frac{\mathrm{4}}{\mathrm{3}}−\mathrm{3}\left({k}+\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{2}} }} \\ $$$$\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}\int\frac{−{dk}}{\sqrt{\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{2}} −\left({k}+\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{2}} }} \\ $$$$=\frac{−\mathrm{1}}{\sqrt{\mathrm{3}}}×{sin}^{−\mathrm{1}} \left(\frac{{k}+\frac{\mathrm{1}}{\mathrm{3}}}{\frac{\mathrm{2}}{\mathrm{3}}}\right) \\ $$$$=\frac{−\mathrm{1}}{\sqrt{\mathrm{3}}}{sin}^{−\mathrm{1}} \left(\frac{\frac{\mathrm{1}}{{t}+\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}}{\frac{\mathrm{2}}{\mathrm{3}}}\right) \\ $$$$\frac{−\mathrm{1}}{\sqrt{\mathrm{3}}}{sin}^{−\mathrm{1}} \left(\frac{\frac{\mathrm{1}}{{x}+\frac{\mathrm{1}}{{x}}+\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}}{\frac{\mathrm{2}}{\mathrm{3}}}\right)+{c} \\ $$$$ \\ $$$$ \\ $$
	Commented by Meritguide1234 last updated on 22/Oct/18

	$${very}\:{nice} \\ $$
	Commented by tanmay.chaudhury50@gmail.com last updated on 22/Oct/18

	$${thank}\:{you}\:{sir}\:... \\ $$
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