calculate-2-2x-3-x-1-3-x-2-x-1-2-dx-

Question Number 81433 by abdomathmax last updated on 13/Feb/20

c a l c u l a t e \int_{2}^{+ \infty} \frac{2 x + 3}{{(x - 1)}^{3} {(x^{2} + x + 1)}^{2}} d x

Answered by MJS last updated on 13/Feb/20

\int \frac{2 x + 3}{{(x - 1)}^{2} {(x^{2} + x + 1)}^{2}} =

Q_{1} (x) = Q_{2} (x) = (x - 1) (x^{2} + x + 1)

P_{1} (x) = - \frac{2}{3} x^{2} - x

P_{2} (x) = - \frac{2}{3} x - 2

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

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

= \frac{4}{3} \ln ∣ x - 1 ∣ - \frac{2}{3} \ln (x^{2} + x + 1) - \frac{2 \sqrt{3}}{3} \arctan \frac{\sqrt{3} (2 x + 1)}{3} =

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

= - \frac{x (2 x + 3)}{3 (x - 1) (x^{2} + x + 1)} - \frac{4}{9} \ln \frac{{(x - 1)}^{2}}{x^{2} + x + 1} + \frac{4 \sqrt{3}}{9} \arctan \frac{\sqrt{3} (2 x + 1)}{3} + C

\Rightarrow

answer is \frac{2 π \sqrt{3}}{9} + \frac{2}{3} - \frac{4}{9} \ln 7 - \frac{4 \sqrt{3}}{9} \arctan \frac{5 \sqrt{3}}{3}

Commented by john santu last updated on 13/Feb/20

Ostrogradski method . i will learn

this method

Commented by abdomathmax last updated on 13/Feb/20

t h a n k y o u s i r m j s