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The cute substitution (Posted on 2019-12-13) Difficulty: 3 of 5
Given that f(x) satisfies the equation f(x)+f(√(1-x2))=2, compute the integral f(x)/√(1-x2) from 0 to 1.

No Solution Yet Submitted by Danish Ahmed Khan    
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Obvious substitution | Comment 1 of 3
An obvious substitution would be x=sin(t), √(1-x^2) = cos(t), so we have
f(sin(t))+f(cos(t)) = 2
which looks a bit nicer. We do not know if f is differentiable, but if we assume it is, when we differentiate both sides, we get
f'(sin(t)) / f'(cos(t)) = sin(t) / cos(t)
to which f'(x) = x is an obvious solution. Hence f(x) = x^2/2 + C. When we plug this into the original equation and solve for C, we get
f(x) = x^2/2 + 3/4
Insert this into the wanted integrand and we get - quite obviously ;) - the result
π/2

  Posted by JLo on 2019-12-14 05:34:55
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