What is the integration of x log x dx ?

Solution :

We have, I = \(\int\) x log x dx

And taking log x as first function and x as second function. Then,

I = log x { \(\int\) x dx } – \(\int\) { \({d\over dx}(log x) \times \int x dx\) } dx

I = (log x) \(x^2\over 2\) – \(\int\) \({1\over x} \times {x^2\over 2}\) dx

\(\implies\) I = \(x^2\over 2\) log x – \(1\over 2\) \(\int\) x dx

\(\implies\) I = \(x^2\over 2\) log x – \(1\over 2\) (\(x^2\over 2\)) + C

\(\implies\) I = \(x^2\over 2\) log x – \(x^2\over 2\) + C

Hence. the integration of x log x with respect to x is \(x^2\over 2\) log x – \(x^2\over 2\) + C