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Question Number 68037 by mathmax by abdo last updated on 03/Sep/19

$$find\int \frac{x^{2}dx}{(x^3-8)(x^4+1)}$$

Answered by MJS last updated on 04/Sep/19

$$\int \frac{x^2}{(x^3-8)(x^4+1)}dx=$$$$$$$$=\frac{1}{51}\int \frac{dx}{x-2}-$$$$-\frac{14}{723}\int \frac{x-\frac{17}{7}}{x^2+2x+4}dx-$$$$-\frac{1-252\sqrt{2}}{8194}\int \frac{x-\frac{8+\sqrt{2}}{62}}{x^2-\sqrt{2}x+1}dx-$$$$-\frac{1+252\sqrt{2}}{8194}\int \frac{x-\frac{8-\sqrt{2}}{62}}{x^2+\sqrt{2}x+1}dx=$$$$$$$$=\frac{1}{51}\ln (x-2)-$$$$-\frac{7}{723}\ln (x^2+2x+4)+\frac{16\sqrt{3}}{723}\arctan \frac{\sqrt{3}(x+1)}{3}-$$$$-\frac{1+252\sqrt{2}}{16388}\ln (x^2-\sqrt{2}x+1)+\frac{2(16-65\sqrt{2})}{4097}\arctan (\sqrt{2}x-1)-$$$$-\frac{1-252\sqrt{2}}{16388}\ln (x^2+\sqrt{2}x+1)-\frac{2(16+65\sqrt{2})}{4097}\arctan (\sqrt{2}x+1)+C$$

Commented by turbo msup by abdo last updated on 04/Sep/19

$$thankyousirmjs$$

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