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Question Number 137290 by JulioCesar last updated on 31/Mar/21

Answered by EDWIN88 last updated on 01/Apr/21

$$\mathrm{Ostrogradski}\mathrm{Integral}$$

Answered by MJS_new last updated on 01/Apr/21

$$\mathrm{Ostrogradski}\mathrm{gives}$$$$-\frac{x^3}{4(x^4+1)}-\frac{1}{4}\int \frac{x^2-4}{x^4+1}dx$$$$\mathrm{this}\mathrm{Integral}\mathrm{decomposed}\mathrm{is}$$$$-\frac{5\sqrt{2}}{32}\int \frac{2x-\sqrt{2}}{x^2-\sqrt{2}x+1}+\frac{3}{16}\int \frac{dx}{x^2-\sqrt{2}x+1}+\frac{5\sqrt{2}}{32}\int \frac{2x+\sqrt{2}}{x^2+\sqrt{2}x+1}+\frac{3}{16}\int \frac{dx}{x^2+\sqrt{2}x+1}$$$$\mathrm{now}\mathrm{solve}\mathrm{with}\mathrm{obvious}\mathrm{formulas}$$

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