Discussion
What is the solution of \(\int x^{2} \ln x\,dx\) ?
Given \(C\) is an arbitrary constant.
Answer: A
The integral is \(\frac{x^{3}}{3}\ln x - \frac{x^{3}}{9} + C\).
Integrate by parts with \(u = \ln x\) and \(dv = x^{2}\,dx\), so \(du = \frac{dx}{x}\) and \(v = \frac{x^{3}}{3}\), writing \(I\) for the integral:
\[ \begin{aligned} I &= \frac{x^{3}}{3}\ln x - \int \frac{x^{2}}{3}\,dx \\ &= \frac{x^{3}}{3}\ln x - \frac{x^{3}}{9} + C \end{aligned} \]
Check by differentiating: \(x^{2}\ln x + \frac{x^{2}}{3} - \frac{x^{2}}{3} = x^{2}\ln x\). The option with \(+\frac{x^{3}}{9}\) drops the minus sign of integration by parts. The options with \(\frac{x^{3}}{9}\ln x\) put the wrong coefficient on the log term.
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