Evaluate ∫24 ln1/2(9 - x)/( ln1/2(9 - x) + ln1/2(x + 3) ) dx.
Offputting, but easy!
The integrand looks horrible. A little playing around, differentiating things like ln1/2(9 - x), soon confirms this impression. But we know the first question in each Putnam session is always easy, so there must be some trick!
We might notice that as x goes from 2 to 4, 9 - x goes from 7 to 5, and x + 3 goes from 5 to 7. That might suggest substituting y such that 9 - y = x + 3 (and hence also y + 3 = 9 - x). dy = - dx, the limits interchange and we get ∫24 ln1/2(y + 3)/( ln1/2(9 - y) + ln1/2(y + 3) ) dy. We are home, because we can add the two integrals to get ∫24 1 dx = 2.
48th Putnam 1987
© John Scholes
7 Oct 1999