calculate-2-dx-x-1-4-x-2-x-1-2-

Question Number 95585 by turbo msup by abdo last updated on 26/May/20

c a l c u l a t e \int_{2}^{+ \infty} \frac{d x}{{(x - 1)}^{4} {(x^{2} + x + 1)}^{2}}

Answered by MJS last updated on 26/May/20

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

[Ostrogradski]

= - \frac{8 x^{4} - 11 x^{3} - 2 x^{2} - x + 9}{27 {(x - 1)}^{4} (x^{2} + x + 1)} - \frac{2}{27} \int \frac{4 x + 5}{(x - 1) (x^{2} + x + 1)} d x

- \frac{2}{27} \int \frac{4 x + 5}{(x - 1) (x^{2} + x + 1)} d x =

= \frac{2}{27} \int \frac{3 x + 2}{x^{2} + x + 1} d x - \frac{2}{9} \int \frac{d x}{x - 1} =

= \frac{2 \sqrt{3}}{81} \arctan \frac{\sqrt{3} (2 x + 1)}{3} + \frac{1}{9} \ln (x^{2} + x + 1) - \frac{2}{9} \ln (x - 1) =

= \frac{2 \sqrt{3}}{81} \arctan \frac{\sqrt{3} (2 x + 1)}{3} + \frac{1}{9} \ln \frac{x^{2} + x + 1}{{(x - 1)}^{2}} + C

\Rightarrow

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

= \frac{13}{63} + \frac{π \sqrt{3}}{81} - \frac{\ln 7}{9} - \frac{2 \sqrt{3}}{81} \arctan \frac{5 \sqrt{3}}{3}

Commented by mathmax by abdo last updated on 27/May/20

thank you sir mjs