Question

Integrate \(x\log x\) with respect to \(x\).

Hint:

Integrate by parts with u = log x, dv = x dx.

Answer 1

Step 1: Choosing parts:
Use integration by parts with \(u=\log x\) (differentiates simply) and \(dv=x\,dx\).

Step 2: Computing du and v:
\(du=\dfrac{1}{x}dx\), \(v=\dfrac{x^2}{2}\).

Step 3: Applying the by-parts formula:
\(\displaystyle\int x\log x\,dx=\dfrac{x^2}{2}\log x-\int\dfrac{x^2}{2}\cdot\dfrac1x\,dx=\dfrac{x^2}{2}\log x-\int\dfrac{x}{2}\,dx\).

Step 4: Finishing the remaining integral:
\(\displaystyle\int\dfrac{x}{2}dx=\dfrac{x^2}{4}\).

Final Answer:
\[ \boxed{\int x\log x\,dx=\dfrac{x^2}{2}\log x-\dfrac{x^2}{4}+C} \]