 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/4/b/solution

Here

I_{0} = x

up to a constant. Substitution of

n = 0

in the integration by parts recurrence gives

I_{1} = x ln x - x + C, x > 0.

Indeed, its derivative is

ln x + 1 - 1 = ln x

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