Question

Folosind metoda schimbarii de variabila sa se calculeza integrala :
Integrala de la radical de 2 la radical de 3 din 2x/(x^4-1) dx

Answer 1

I=∫2xdx(x^4-1)  x∈[√2;√3] face  substitutia x²=y  2xdx=dy
se recvalculeaza  limitele  de  integrare pt  x=√2  =>y=2  x=√3  y=3
integrala  devine
I=∫dy/(y²-1)=∫dy/(y+1)(y-1)  unde  y∈[2,3]
1/(y-1)(y+1)=A/(y-1)+B/(y+1)=(Ay+A+BY-B)/(y-1)(y+1)=[(A+B)y+(A-B)]/))Y-1)(y+1)
Folosim  metoda  identificarii
{A+B=0
{A-B=1  =>A=1/2  B=- 1/2
Integrala  devine
I=1/2∫dy/(y+1)-1/2∫dy/(y-1)=1/2[ln(y+1)-ln(y-1)]=1/2ln(y+1)/(y-1)  y∈[2.3]
I=ln(3+1)/(3-1)-ln(2+1)/(2-1)=ln2-ln3=ln2/3