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Today's calculation of Integral 209

Source: Yokohama National University entrance exam/Engineering 1972

June 11, 2007
calculusintegrationtrigonometrycalculus computations

Problem Statement

Let m, nm,\ n be the given distinct positive integers. Answer the following questions. (1) Find the real number α (∣α∣<1)\alpha \ (|\alpha |<1) such that ∫−ππsin⁡(m+α)x sin⁡(n+α)x dx=0\int_{-\pi}^{\pi}\sin (m+\alpha )x\ \sin (n+\alpha )x\ dx=0. (2) Find the real number β\beta satifying the sytem of equation ∫−ππsin⁡2(m+β)x dx=π+24m−1\int_{-\pi}^{\pi}\sin^{2}(m+\beta )x\ dx=\pi+\frac{2}{4m-1}, ∫−ππsin⁡2(n+β)x dx=π+24n−1\int_{-\pi}^{\pi}\sin^{2}(n+\beta )x\ dx=\pi+\frac{2}{4n-1}.
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