Intégration des fonctions rationnelles - Exemple

Calcul de \(\displaystyle{\int_0^1\frac{dt}{(t+1)(t^2+1)}}\):

On a : \(\displaystyle{2\int_0^1\frac{dt}{(t+1)(t^2+1)}=\int_0^1\frac{dt}{(t+1)}-\int_0^1\frac{tdt}{(t^2+1)}+\int_0^1\frac{dt}{(t^2+1)}}\), soit

\(\displaystyle{2\int_0^1\frac{dt}{(t+1)(t^2+1)}=[\ln(t+1)]_0^1-\frac{1}{2}[\ln(t^2+1)]_0^1+[\arctan t]_0^1}\)

Donc, finalement

\(\displaystyle{\int_0^1\frac{dt}{(t+1)(t^2+1)}=\frac{\pi}{8}+\frac{1}{4}\ln2}\)

Le calcul avec Maple

> Int(1/((x+1)*(x^2+1)),x);

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

> int(1/((x+1)*(x^2+1)),x);

\(\displaystyle{\frac{1}{2}\ln(x+1)-\frac{1}{4}\ln(x^2+1)+\frac{1}{2}\arctan(x)}\)

Entre 0 et 1

> int(1/((x+1)*(x^2+1)),x=0..1);

\(\displaystyle{\frac{1}{4}\ln(2)+\frac{1}{8}\pi}\)