Question

f:(0,+infinit) -> R , f(x)= x lnx
F:(0,+infinit)->R , F(x)= x^2/2 (lnx-1/2)

Sa se arate ca F este o primitiva a lui f.

Answer 1

F(x) este primitiva lui f(x) daca F'(x)=f(x)

$F(x)= \frac{1}{2}(x^2lnx- \frac{x^2}{2})= \frac{1}{2}x^2lnx- \frac{1}{4}x^2 \\ F'(x)= \frac{1}{2}( 2xlnx+ \frac{x^2}{x})- \frac{1}{4}2x= \\ \\ F'(x)=xlnx+ \frac{x}{2}- \frac{x}{2}=xlnx$