MathDB
Today's Calculation of Integral 47

Source: 2001 Tokyo University

June 17, 2005
calculusintegrationfunctiontrigonometrycalculus computations

Problem Statement

Find the condition of a,ba,b for which the function f(x) (0≤x≤2π)f(x)\ (0\leq x\leq 2\pi) satisfying the following equality can be determined uniquely,then determine f(x)f(x), assuming that f(x)f(x) is a continuous function at 0≤x≤2π0\leq x\leq 2\pi. f(x)=a2π∫02πsin⁡(x+y)f(y)dy+b2π∫02πcos⁡(x−y)f(y)dy+sin⁡x+cos⁡xf(x)=\frac{a}{2\pi}\int_0^{2\pi} \sin (x+y)f(y)dy+\frac{b}{2\pi}\int_0^{2\pi} \cos (x-y)f(y)dy+\sin x+\cos x
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