Question

imi cere sa arat ca I2 = 1/3 ln2

Answer 1

$\int\limits^1_0 {x^{2}/ x^{3}+1 } \, dx =1/3* \int\limits^1_0 {3 x^{2} / x^{3}+1 } \, dx= 1/3*\int\limits^1_0 {(x^{3}+1)'/ x^{3}+1 } \, dx$
$=1/3*ln( x^{3}+1 ) | _{0} ^{1} =1/3*ln(1+1)-1/3*ln(0+1)=$
$=1/3*ln 2-1/3*ln1=1/3*ln2$