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Today's calculation of Integral 878

Source: 2013 National Defense Academy of Japan entrance exam

June 29, 2013
calculusintegrationfunctiontrigonometrycalculus computations

Problem Statement

A cubic function f(x)f(x) satisfies the equation sin⁡3t=f(sin⁡t)\sin 3t=f(\sin t) for all real numbers tt.
Evaluate ∫01f(x)21−x2 dx\int_0^1 f(x)^2\sqrt{1-x^2}\ dx.
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