Question

Integrala de la 0 la a din (2x+1)/(x^2+x+1)dx=ln3
Cat e a? Si cum se demonstreaza? a>0

Answer 1

∫ (2x+1)/(x²+x+1)dx= ∫ (x²+x+1)' /(x²+x+1) dx =  ∫[ln(x²+x+1)]' dx = ln(x²+x+1) luat de la 0 la a = ln(a²+a+1) - ln(0²+0+1)= ln(a²+a+1) - ln1 = ln(a²+a+1)= ln3    
a²+a+1=3      a²+a-2=0       (a+2)(a-1)=0    a1=-2 si a2=1   solutie a=1