[tex]\text{Prin alte cuvinte problema ne cere sa calculam }\displaystyle \int_{1}^ef(x)dx\\
\text{Calculam integrala folosim integrarea prin parti :}\\
\int _{1}^ex\cdot \ln xdx=\int_1^e \left(\dfrac{x^2}{2}\right)'\cdot lnxdx=\dfrac{x^2}{2}\ln x|_1^e-\int_1^e \dfrac{x^2}{2}\cdot \dfrac{1}{x}dx=\\
\dfrac{e^2}{2}\cdot \ln e-\dfrac{1}{2}\cdot \ln 1-\dfrac{x^2}{4}|_1^e=\dfrac{e^2}{2}-\dfrac{e^2}{4}+\dfrac{1}{4}=\dfrac{e^2+1}{4} (\text{adica f)})[/tex]
