The Calculus of Friendship by Strogatz Steven

Author:Strogatz, Steven
Language: eng
Format: epub
ISBN: 9781400830886
Publisher: Princeton University Press

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Now let’s turn to the integral

that you mentioned came up when you tried to calculate the period of the cubic oscillator by using conservation of energy. As I guessed (and you did too), it’s an elliptic integral. I’ll explain those some other time. For now let’s just see what one can do to get a number out.

First, notice the integral is finite since 1—x2 ≤ 1—x4 for 0 ≤ x ≤ 1, and so

So we know

Next, I resorted to looking in the mathematical tables by Abramowitz & Stegun (the Bible of such stuff). There in Chapter 17, Elliptic Integrals, I found Formula 17.4.56, which says essentially

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