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Today's calculation of Integral 187

Source: Tsukuba university entrance exam 2007

March 1, 2007
calculusintegrationtrigonometryinequalitiescalculus computations

Problem Statement

For a constant a,a, let f(x)=axsin⁡x+x+π2.f(x)=ax\sin x+x+\frac{\pi}{2}. Find the range of aa such that ∫0π{f′(x)}2 dx≥f(π2).\int_{0}^{\pi}\{f'(x)\}^{2}\ dx \geq f\left(\frac{\pi}{2}\right).
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