2.56: Stationary point and integration using a derivative

0606/22/O/N/19 — Question 10 · 11 marks

(i)
Given that y=ln⁡xx2y = \dfrac{\ln x}{x^{2}}, find dydx\dfrac{\mathrm{d}y}{\mathrm{d}x}.
[3]
(ii)
Find the coordinates of the stationary point on the curve y=ln⁡xx2y = \dfrac{\ln x}{x^{2}}.
[3]
(iii)
Using your answer to part (i), find ∫ln⁡xx3 dx\displaystyle\int \frac{\ln x}{x^{3}}\,\mathrm{d}x.
[3]
(iv)
Hence evaluate ∫12ln⁡xx3 dx\displaystyle\int_{1}^{2} \frac{\ln x}{x^{3}}\,\mathrm{d}x.
[2]