Question

Calculati integrala de la 2 la 3 din : 3/(x^2 + x - 2 ) . Va rog mult sa ma ajutati. Multumesc!

Answer 1

∫
x²+x-2=(x-1)(x+2)=>
3/(x²+x-2)=3/(x-1)(x+2)
3/(x-1)(x+2)=A/(x-1)+B/(x+2)=[A(x+1)+B(x-2)]/(x-1)(x+2)=
(Ax+BX+2A-b)/(x-1)(x+2)
x(A+B)+2A-B=3
Se  rezolva prin  identificare
{A+b=0  B=-A
{2A-B=3  2A+A=3=> A=1  => B=-1
Integrala devine
I=∫3dx/(x²+x-2)=∫dx/(x-1)-∫dx/(x+2)=ln(x-1)-ln(x+2)=ln(x-1)/(x+2)/2↑3=ln2/5-ln1/4=
ln2/5:1/4=ln2/5*4=ln8/5
I=