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If the integral from 0 to x/(1+x²) of f(t) dt = x for all x > 0, find 72*f(20).

1. 1
2. 2
3. 3
4. 4

Correct answer: 2

Solution

Differentiating: f(x/(1+x²)) * d/dx[x/(1+x²)] = 1. d/dx[x/(1+x²)] = (1-x²)/(1+x²)². So f(x/(1+x²)) = (1+x²)²/(1-x²). Since x/(1+x²) <= 1/2 for all x > 0 (by AM-GM), the value x=20 is outside the range of the upper limit. The problem likely intends a different upper limit or the answer 72*f(20)=2 is given directly.

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