The Inevitable π

Cambridge STEP 1 | 1987 | Question 5

Hello my dear friends of Math Games, welcome back to the 5th question of 1987’s Cambridge STEP Exam. As a reminder, these questions were designed to test out the most mathematically capable students hoping to read mathematics at the University of Cambridge.

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What good is a math problem if not to be solved with grit and wit? What follows is a complete solution, but a complete solution, read cold, teaches rather less than a partial attempt, wrestled with honestly.

Ergo, set aside a few minutes. Work through the steps. Let the difficulty be felt. The argument will mean considerably more once you have traced some part of it yourself.

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Solution

First we let psi be the give integral.

Then if we let

Then we can say

Also, we have

Substituting all of these results into the integral, and changing limits as required, we reach the result:

This simplifies nicely.

And that’s the first part done!

Now let’s find out what happens when alpha > beta.