x-1-x-2-4x-5-2-dx-

Question Number 92055 by 675480065 last updated on 04/May/20

\int (\frac{x + 1}{{(x^{2} + 4 x + 5)}^{2}}) dx

Commented by Prithwish Sen 1 last updated on 04/May/20

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

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

= \frac{- 1}{2 (x^{2} + 1)} - \frac{x}{2 (1 + x^{2})} - \frac{1}{2} \tan^{- 1} x + c

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

I just forgot to do the inverse transformation .

thank you . please check .

Commented by 675480065 last updated on 04/May/20

i dont get u

Commented by Prithwish Sen 1 last updated on 04/May/20

\frac{x + 1}{{(x^{2} + 4 x + 5)}^{2}} and \frac{x - 1}{{(x^{2} + 1)}^{2}} are the same function

only the function has been a horizontal shifting

along x axis in the rightward direction . There

should not be any impact on overall calculation .