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