Evaluate the integral \(\int x \ln x d x\) [solved]

Question

Evaluate the integral \(\int x \ln x d x\)

Answer 1

\(\int x \ln x d x\)
\(u=\ln x\)
\(d v=x d x\)
\(d u=\frac{1}{x} d x\)
\(v=\frac{x^{2}}{2}\)
\(\int x \ln s d x=\frac{x^{2}}{2} \ln x-\int \frac{x^{2}}{2} \cdot \frac{1}{x} d x\)
\(\frac{x^{2}}{2} \ln x-\frac{1}{2} \int x d x\)
\(\frac{x^{2}}{2} \ln x-\frac{1}{2}\left(\frac{x^{2}}{2}\right)+c\)
\(\frac{x^{2}}{2} \ln x-\frac{1}{4} x^{2}+c\)