1-0-1-2-2x-x-2-3-dx-Improper-integral-

Question Number 211696 by mnjuly1970 last updated on 16/Sep/24

{\int_{0}}^{1} \frac{1}{{(2 + 2 x + x^{2})}^{3}} d x = ?

\underset{⏟}{\underset{\overset{Improper integral}{}}{}}

- - - - - - - - -

Answered by Ghisom last updated on 16/Sep/24

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

[{Ostrogradski}^{'} s Method]

= \frac{(x + 1) (3 x^{2} + 6 x + 8)}{8 {(x^{2} + 2 x + 2)}^{2}} + \frac{3}{8} \int \frac{d x}{x^{2} + 2 x + 2} =

= \frac{(x + 1) (3 x^{2} + 6 x + 8)}{8 {(x^{2} + 2 x + 2)}^{2}} + \frac{3}{8} \arctan (x + 1) + C

\Rightarrow

\underset{0}{\int^{1}} \frac{d x}{{(x^{2} + 2 x + 2)}^{3}} = - \frac{2}{25} + \arctan \frac{1}{3}

Facebook Tweet Pin