What is the integration of xlogx?

Answer 1

The integral of ( x \log x ) can be found using integration by parts. Letting ( u = \log x ) and ( dv = x , dx ), we find ( du = \frac{1}{x} , dx ) and ( v = \frac{x^2}{2} ). Applying integration by parts yields:

[ \int x \log x , dx = \frac{x^2}{2} \log x - \int \frac{x^2}{2} \cdot \frac{1}{x} , dx = \frac{x^2}{2} \log x - \frac{x^2}{4} + C ]

Thus, the result is:

[ \int x \log x , dx = \frac{x^2}{2} \log x - \frac{x^2}{4} + C ]