 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2007/paper-1/section-a/3/b/solution

u = ln x

and

d v = x d x

gives

\int x ln x d x = \frac{x ^{2}}{2} ln x - \frac{x ^{2}}{4} + C, x > 0.

The remaining integral is

\frac{1}{2} \int x d x

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