1.12: Partial fractions and logarithmic integration

0606/11/M/J/20 — Question 8 · 9 marks

(a)
Show that 32x−3+32x+3\dfrac{3}{2x - 3} + \dfrac{3}{2x + 3} can be written as 12x4x2−9\dfrac{12x}{4x^{2} - 9}.
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(b)
Hence find ∫12x4x2−9 dx\displaystyle\int \frac{12x}{4x^{2} - 9}\,\mathrm{d}x, giving your answer as a single logarithm and an arbitrary constant.
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(c)
Given that ∫2a12x4x2−9 dx=ln⁡55\displaystyle\int_{2}^{a} \frac{12x}{4x^{2} - 9}\,\mathrm{d}x = \ln 5\sqrt{5}, where a>2a > 2, find the exact value of aa.
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