1.71: Logarithmic differentiation and definite integration

0606/13/O/N/16 — Question 6 · 7 marks

(i)
Find ddx(ln⁡(3x2−11))\dfrac{\mathrm{d}}{\mathrm{d}x}(\ln(3x^{2} - 11)).
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(ii)
Hence show that ∫x3x2−11 dx=pln⁡(3x2−11)+c\displaystyle\int \frac{x}{3x^{2} - 11}\,\mathrm{d}x = p\ln(3x^{2} - 11) + c, where pp is a constant to be found and cc is a constant of integration.
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(iii)
Given that ∫2ax3x2−11 dx=ln⁡2\displaystyle\int_{2}^{a} \frac{x}{3x^{2} - 11}\,\mathrm{d}x = \ln 2, where a>2a > 2, find the value of aa.
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