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Today's calculation of Integral 812

Source: 2012 Kwansei　Gakuin University entrance exam/Science

April 11, 2012
calculusintegrationtrigonometrylimitlogarithmscalculus computations

Problem Statement

Let f(x)=cos⁡2x−(a+2)cos⁡x+a+1sin⁡x.f(x)=\frac{\cos 2x-(a+2)\cos x+a+1}{\sin x}. For constant aa such that lim⁡x→0f(x)x=12\lim_{x\rightarrow 0} \frac{f(x)}{x}=\frac 12, evaluate ∫π3π21f(x)dx.\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{f(x)}dx.
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