calculer-la-primitive-de-t-2-1-t-2-2-dt-

Question Number 166182 by SANOGO last updated on 14/Feb/22

c a l c u l e r l a p r i m i t i v e d e

\int \frac{t^{2}}{{(1 + t^{2})}^{2}} d t

Answered by greogoury55 last updated on 14/Feb/22

t = \tan u

Y = \int \frac{\tan^{2} u}{{(1 + \tan^{2} u)}^{2}} \sec^{2} u d u

Y = \int \frac{\tan^{2} u}{\sec^{4} u} \sec^{2} u d u

Y = \int \frac{\tan^{2} u}{\sec^{2} u} d u = \int \sin^{2} u d u

Y = \int \frac{1 - \cos 2 u}{2} d u = \frac{u - \frac{1}{2} \sin 2 u}{2} + c

Y = \frac{1}{2} \arctan (t) - \frac{t}{2 t^{2} + 2} + c

Commented by SANOGO last updated on 14/Feb/22

m e r c i

Answered by MJS_new last updated on 15/Feb/22

\int \frac{t^{2}}{{(t^{2} + 1)}^{2}} d t =

[{Ostrogradski}^{'} s Method]

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

= - \frac{t}{2 (t^{2} + 1)} + \frac{1}{2} \arctan t + C

Facebook Tweet Pin