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Question Number 165853 by mathlove last updated on 09/Feb/22

Answered by MJS_new last updated on 09/Feb/22

$$\int \frac{2x-1}{{(x+2)}^4{(x-1)}^4}dx=$$$$\left[{\mathrm{Ostrogradski}}^{\prime }\mathrm{s}\mathrm{Method}\right]$$$$=\frac{40x^5+100x^4-140x^3-310x^2+278x-49}{729{(x-1)}^3{(x+2)}^3}+\frac{40}{729}\int \frac{dx}{(x-1)(x+2)}=$$$$[\int \frac{dx}{(x-1)(x+2)}=\frac{1}{3}\int (\frac{1}{x-1}-\frac{1}{x+2})dx=\frac{1}{3}\ln \mid \frac{x-1}{x+2}\mid ]$$$$==\frac{40x^5+100x^4-140x^3-310x^2+278x-49}{729{(x-1)}^3{(x+2)}^3}+\frac{40}{2187}\ln \mid \frac{x-1}{x+2}\mid +C$$

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