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Today's calculation of Integral 159

Source: Kyoto University entrance exam/science 2000 2nd round

October 7, 2006
calculusintegrationfunctiongeometrytrigonometrycalculus computations

Problem Statement

A function is defined by f(x)=∫0x11+t2dt.f(x)=\int_{0}^{x}\frac{1}{1+t^{2}}dt. (1) Find the equation of normal line at x=1x=1 of y=f(x).y=f(x). (2) Find the area of the figure surrounded by the normal line found in (1), the xx axis and the graph of y=f(x).y=f(x). Note that you may not use the formula ∫11+x2dx=tan⁡−1x+Const.\int \frac{1}{1+x^{2}}dx=\tan^{-1}x+Const.
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