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Today's calculation of Integral 877

Source: 2005 Tokyo Gakugei University entrance exam/Education

June 17, 2013
calculusintegrationlimittrigonometrycalculus computationsDefinite integrallimits

Problem Statement

Let f(x)=lim⁡n→∞sin⁡n+2x+cos⁡n+2xsin⁡nx+cos⁡nxf(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x} for 0≤x≤π2.0\leq x\leq \frac{\pi}2.
Evaluate ∫0π2f(x) dx.\int_0^{\frac{\pi}2} f(x)\ dx.
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