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Today's calculation of Integral 860

Source: 2013 Kyorin University entrance exam/Medicine

January 24, 2013
calculusintegrationfunctionlogarithmsanalytic geometrygeometryderivative

Problem Statement

For a function f(x) (x≥1)f(x)\ (x\geq 1) satisfying f(x)=(log⁡ex)2−∫1ef(t)tdtf(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt, answer the questions as below.
(a) Find f(x)f(x) and the yy-coordinate of the inflection point of the curve y=f(x)y=f(x).
(b) Find the area of the figure bounded by the tangent line of y=f(x)y=f(x) at the point (e, f(e))(e,\ f(e)), the curve y=f(x)y=f(x) and the line x=1x=1.
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