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Question Number 130138 by mathmax by abdo last updated on 22/Jan/21
1) decompose F(x)=((x^2 −3)/((x^2 −1)^2 (x^2 +4)^3 )) 2) determine ∫ F(x)dx
$$\left.\mathrm{1}\right)\:\mathrm{decompose}\:\mathrm{F}\left(\mathrm{x}\right)=\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{3}}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{x}^{\mathrm{2}} +\mathrm{4}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{2}\right)\:\mathrm{determine}\:\int\:\mathrm{F}\left(\mathrm{x}\right)\mathrm{dx} \\ $$
Answered by MJS_new last updated on 23/Jan/21
decomposition ∫((x^2 −3)/((x^2 −1)^2 (x^2 +4)^3 ))dx= =∫(−(1/(250(x−1)^2 ))+(8/(625(x−1)))−(1/(250(x+1)^2 ))−(8/(625(x+1)))−(7/(25(x^2 +4)^3 ))−(9/(125(x^2 +4)^2 ))−((11)/(625(x^2 +4))))dx ... Ostrogradski ∫((x^2 −3)/((x^2 −1)^2 (x^2 +4)^3 ))dx= =−((x(121x^4 +3x^2 −3324))/(16000(x^2 −1)(x^2 +4)^2 ))−(1/(16000))∫((121x^2 −2169)/((x^2 −1)(x^2 +4)))dx= =...+(8/(625))∫(dx/(x−1))−(8/(625))∫(dx/(x+1))−((2653)/(80000))∫(dx/(x^2 +4))= ... =−((x(121x^4 +3x^2 −3324))/(16000(x^2 −1)(x^2 +4)^2 ))+(8/(625))ln ∣((x−1)/(x+1))∣ −((2653)/(160000))arctan (x/2) +C
$$\mathrm{decomposition} \\ $$$$\int\frac{{x}^{\mathrm{2}} −\mathrm{3}}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} \left({x}^{\mathrm{2}} +\mathrm{4}\right)^{\mathrm{3}} }{dx}= \\ $$$$=\int\left(−\frac{\mathrm{1}}{\mathrm{250}\left({x}−\mathrm{1}\right)^{\mathrm{2}} }+\frac{\mathrm{8}}{\mathrm{625}\left({x}−\mathrm{1}\right)}−\frac{\mathrm{1}}{\mathrm{250}\left({x}+\mathrm{1}\right)^{\mathrm{2}} }−\frac{\mathrm{8}}{\mathrm{625}\left({x}+\mathrm{1}\right)}−\frac{\mathrm{7}}{\mathrm{25}\left({x}^{\mathrm{2}} +\mathrm{4}\right)^{\mathrm{3}} }−\frac{\mathrm{9}}{\mathrm{125}\left({x}^{\mathrm{2}} +\mathrm{4}\right)^{\mathrm{2}} }−\frac{\mathrm{11}}{\mathrm{625}\left({x}^{\mathrm{2}} +\mathrm{4}\right)}\right){dx} \\ $$$$… \\ $$$$ \\ $$$$\mathrm{Ostrogradski} \\ $$$$\int\frac{{x}^{\mathrm{2}} −\mathrm{3}}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} \left({x}^{\mathrm{2}} +\mathrm{4}\right)^{\mathrm{3}} }{dx}= \\ $$$$=−\frac{{x}\left(\mathrm{121}{x}^{\mathrm{4}} +\mathrm{3}{x}^{\mathrm{2}} −\mathrm{3324}\right)}{\mathrm{16000}\left({x}^{\mathrm{2}} −\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{4}\right)^{\mathrm{2}} }−\frac{\mathrm{1}}{\mathrm{16000}}\int\frac{\mathrm{121}{x}^{\mathrm{2}} −\mathrm{2169}}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{4}\right)}{dx}= \\ $$$$=…+\frac{\mathrm{8}}{\mathrm{625}}\int\frac{{dx}}{{x}−\mathrm{1}}−\frac{\mathrm{8}}{\mathrm{625}}\int\frac{{dx}}{{x}+\mathrm{1}}−\frac{\mathrm{2653}}{\mathrm{80000}}\int\frac{{dx}}{{x}^{\mathrm{2}} +\mathrm{4}}= \\ $$$$… \\ $$$$=−\frac{{x}\left(\mathrm{121}{x}^{\mathrm{4}} +\mathrm{3}{x}^{\mathrm{2}} −\mathrm{3324}\right)}{\mathrm{16000}\left({x}^{\mathrm{2}} −\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{4}\right)^{\mathrm{2}} }+\frac{\mathrm{8}}{\mathrm{625}}\mathrm{ln}\:\mid\frac{{x}−\mathrm{1}}{{x}+\mathrm{1}}\mid\:−\frac{\mathrm{2653}}{\mathrm{160000}}\mathrm{arctan}\:\frac{{x}}{\mathrm{2}}\:+{C} \\ $$
Commented by mathmax by abdo last updated on 23/Jan/21
thank you sir.
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir}. \\ $$
Commented by Lordose last updated on 23/Jan/21
Sir is there any software capable of solving the variables in applying ostrograski
$$\mathrm{Sir}\:\mathrm{is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{software}\:\mathrm{capable}\:\mathrm{of}\:\mathrm{solving} \\ $$$$\mathrm{the}\:\mathrm{variables}\:\mathrm{in}\:\mathrm{applying}\:\mathrm{ostrograski}\: \\ $$
Commented by MJS_new last updated on 23/Jan/21
I don′t know...
$$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{know}… \\ $$

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