I-dx-x-8-x-6-J-1-x-7-x-1-x-7-dx-Calculate-I-and-J-

Question Number 165561 by LEKOUMA last updated on 03/Feb/22

I = \int \frac{d x}{x^{8} + x^{6}}

J = \int \frac{1 - x^{7}}{x (1 + x^{7})} d x

C a l c u l a t e I a n d J

Answered by MJS_new last updated on 05/Feb/22

\int \frac{d x}{x^{8} + x^{6}} = \int (\frac{1}{x^{6}} - \frac{1}{x^{4}} + \frac{1}{x^{2}} - \frac{1}{x^{2} + 1}) d x =

= - \frac{1}{5 x^{5}} + \frac{1}{3 x^{3}} - \frac{1}{x} - \arctan x =

= - \frac{15 x^{4} - 5 x^{2} + 3}{15 x^{5}} - \arctan x + C

Answered by MJS_new last updated on 05/Feb/22

\int \frac{1 - x^{7}}{x (1 + x^{7})} d x = - \int \frac{x^{7} - 1}{x (x + 1) (x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1)} d x =

= \int (\frac{1}{x} - \frac{2}{7 (x + 1)} - \frac{2 (6 x^{5} - 5 x^{4} + 4 x^{3} - 3 x^{2} + 2 x - 1)}{7 (x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1)}) d x =

= \ln x - \frac{2}{7} \ln (x + 1) - \frac{2}{7} \ln (x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1) =

= \ln ∣ x ∣ - \frac{2}{7} \ln ∣ x^{7} + 1 ∣ + C

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