1/((1+y).(1+yx²)) = A/(1+y) + B/(1+yx²)
A.(1+yx²) + B(1+y) = 1
(A+B) + y(Ax²+B) = 1
A+B = 1
Ax²+B = 0
A - Ax² = 1
A = 1/(1-x²)
B = 1 - 1/(1-x²) = (1-x²-1)/(1-x²) = -x²/(1-x²)
***
1/((1+y).(1+yx²)) = ( 1/(1-x²))/(1+y) - (x²/(1-x²))/(1+yx²)
S 1/((1+y).(1+yx²)) dx = ( 1/(1-x²)).ln|1+y| - (x²/(1-x²)) * ln|1+yx²| * 1/x²
S 1/((1+y).(1+yx²)) dx = ( 1/(1-x²)).ln|1+y| - (1/(1-x²)).ln|1+yx²|
S 1/((1+y).(1+yx²)) dx = ( 1/(1-x²)).(ln|1+y| - ln|1+yx²|)
S 1/((1+y).(1+yx²)) dx = ( 1/(1-x²)).(ln|(1+y)/(1+yx²)|)
S(de 0 à +oo) 1/((1+y).(1+yx²)) dx = ( 1/(1-x²)) * [ln|(1+y)/(1+yx²)|](de0à+oo)
S(de 0 à +oo) 1/((1+y).(1+yx²)) dx = ( 1/(1-x²)) * [ln(1/x²) - ln(1)]
S(de 0 à +oo) 1/((1+y).(1+yx²)) dx = ( 1/(1-x²)) * ln(1/x²) = (1/(x²-1)) * ln(x²) = 2.ln|x|/(x²-1)
Et si on se limite à x sur R*+ et différent de 1, on peut écrire S(de 0 à +oo) 1/((1+y).(1+yx²)) dx = 2.ln(x)/(x²-1)