Edexcel IALGeneralGeneral Applications8 Marks
(a) Show that `d/dx ((ln x)/x) = (1 - ln x)/(x²)`. (b) Find the exact value of `∫₁^(e) (1 - ln x)/(x²) dx`.
- (a)Show that `d/dx ((ln x)/x) = (1 - ln x)/(x²)`.[3]
- (b)Find the exact value of `∫₁^(e) (1 - ln x)/(x²) dx`.[5]
Mark scheme
Marking points from the official mark scheme. The mark or mark type is shown in brackets.
- M1 for using the quotient rule `(vdu/dx - u dv/dx)/v^2` (or product rule `u'v+uv'` for `ln x * x^-1`).
- A1 for `u=ln x, du/dx=1/x` and `v=x, dv/dx=1` (or `v=x⁻¹, dv/dx=-x⁻²`).
- A1 for `(x(1/x) - ln x(1)) / x² = (1 - ln x) / x²` (cso).
- M1 for recognizing that `∫ (1 - ln x)/(x²) dx = (ln x)/x` (from part a).
- B1 for correct limits.
- M1 for substituting limits into `[(ln x)/x]₁^(e)`.
- A1 for `(ln e)/e - (ln 1)/1` or `1/e - 0`.
- A1 for `1/e`.
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