Show that the area bounded by the curve y = x l n x − c , the x-axis and the vertical line through the maximum point of the curve is independent of the constant c.

Question

Show that the area bounded by the curve y = x l n x − c ​ , the x-axis and the vertical line through the maximum point of the curve is independent of the constant c.

TestHub · Accepted Answer

Differentiate to locate the maximum, then substitute u=ln x-c to evaluate the area: y ′ = x 2 1 − ln x + c ​ . x max ​ = e c + 1 , x zero ​ = e c . A = ∫ e c e c + 1 ​ x ln x − c ​ d x = ∫ 0 1 ​ u d u = 2 1 ​ . ​

Mathematics - Area Under Curves Question with Solution | TestHub

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