 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/17/d/solution

Let

H (u) = sin^{2} u + ln u

. The chain rule gives

H^{'} (u) = sin 2 u + 1/ u

, so the two given summands combine into its logarithmic derivative,

f (u) = H^{'} (u) / H (u)

. For

u > 1

H (u) > 0

, so the real logarithm is defined throughout the integration path. Since

H (π) = ln π

F (x) = ln (\frac{sin ^{2} x + ln x}{ln π}), x > 1.

This satisfies

F (π) = 0

, and differentiation recovers both terms of the integrand.

 New to topics? Read the docs here!