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Question Number 130138 by mathmax by abdo last updated on 22/Jan/21

$$1)\mathrm{decompose}\mathrm{F}\left(\mathrm{x}\right)=\frac{{\mathrm{x}}^2-3}{{({\mathrm{x}}^2-1)}^2{({\mathrm{x}}^2+4)}^3}$$$$2)\mathrm{determine}\int \mathrm{F}\left(\mathrm{x}\right)\mathrm{dx}$$

Answered by MJS_new last updated on 23/Jan/21

$$\mathrm{decomposition}$$$$\int \frac{x^2-3}{{(x^2-1)}^2{(x^2+4)}^3}dx=$$$$=\int (-\frac{1}{250{(x-1)}^2}+\frac{8}{625(x-1)}-\frac{1}{250{(x+1)}^2}-\frac{8}{625(x+1)}-\frac{7}{25{(x^2+4)}^3}-\frac{9}{125{(x^2+4)}^2}-\frac{11}{625(x^2+4)})dx$$$$...$$$$$$$$\mathrm{Ostrogradski}$$$$\int \frac{x^2-3}{{(x^2-1)}^2{(x^2+4)}^3}dx=$$$$=-\frac{x(121x^4+3x^2-3324)}{16000(x^2-1){(x^2+4)}^2}-\frac{1}{16000}\int \frac{121x^2-2169}{(x^2-1)(x^2+4)}dx=$$$$=...+\frac{8}{625}\int \frac{dx}{x-1}-\frac{8}{625}\int \frac{dx}{x+1}-\frac{2653}{80000}\int \frac{dx}{x^2+4}=$$$$...$$$$=-\frac{x(121x^4+3x^2-3324)}{16000(x^2-1){(x^2+4)}^2}+\frac{8}{625}\ln \mid \frac{x-1}{x+1}\mid -\frac{2653}{160000}\arctan \frac{x}{2}+C$$

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

$$\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{know}...$$

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